Category Archives: calculus

Limits and Continuity: Part 10: Tutorial Problems for IITJEE Maths

Problem 1:

The point of discontinuity of the function:

f(x) = \frac{1}{\sin{x} - \cos{x}} in the closed interval [0, \frac{\pi}{2}] are:

(a) 0 and \frac{\pi}{2} (b) \frac{\pi}{2} and \frac{\pi}{4}

(c) \frac{\pi}{4} and 0 (d) \frac{\pi}{4}

Problem 2:

Given f(x) = \frac{x^{2}-9}{x-3} for 0 \leq x <3 and f(x) = 4x-5 for 3 \leq x \leq 6

Consider:

(i) f(x) is discontinuous in (0,3)

(ii) f(x) is discontinuous in (3,6)

(iii) f(x) is continuous in [0,6]

(iv) \lim_{x \rightarrow 3} f(x) exists.

Which of the above statements are false?

(a) only 1 and 2 (b) only 2 and 4 (c) only 1 and 3 (d) none of a, b, c

Problem 3:

For the following function:

f(x) = \frac{x^{2}-3x+2}{x-3} for 0 \leq x \leq 4, and

f(x) = \frac{x^{2}+1}{x-2} for 4 < x \leq 6

Consider

(i) f(x) is discontinuous in (0,4)

(ii) f(x) is discontinuous in (4,6)

(iii) f(x) is discontinuous in [0,6]

(iv) \lim_{x \rightarrow 3}f(x) exists

Which of the above statements are true?

(a) only 1 and 2 (b) only 2 and 4 (c) only 1 and 3 (d) none of a, b and c

Problem 4:

If the function f(x) where

f(x) = \frac{(3^{x}-1)^{2}}{\tan{x} \log{(1+x)}} for x \neq 0

f(x) = \log{k} . \log{\sqrt{3}} for x=0

is continuous at x=0, then k=

(a) 6 (b) \sqrt{3} (c) 9 (d) \frac{3}{2}

Problem 5:

At x = \frac{3 \pi}{4}, the function f(x) where

\frac{\cos{x} + \sin{x}}{3\pi -4x} , where x \neq \frac{3\pi}{4}

f(x) = \frac{1}{\sqrt{2}} where x = \frac{3\pi}{4}

has

(a) removable discontiuity

(b) irremovable discontinuity

(c) no discontinuity

(d) none

Problem 6:

If f(x) is given to be continuous at x=0, where

f(x) = \frac{(e^{kx}-1) \sin{(kx)}}{x^{2}} for x \neq 0 and f(0)=4, then the value of k is:

(a) 2 (b) -2 (c) \pm {2} (d) \pm {\sqrt{2}}

Problem 7:

If given function f(x) is continuous at zero and if

f(x) = \frac{4^{x}-2^{x+1}+1}{1-\cos{x}} when x \neq 0 and f(0)=k, then the value of k is :

(a) \frac{1}{2}(\log{2})^{2} (b) 2(\log{2})^{2} (c) 4 \log{2} (d) \frac{1}{4} \log{2}

Problem 8:

If f(x) is continuous at x=3, where

f(x) = \frac{(2^{x}-8) \log{(x-2)}}{1- \cos{(x-3)}} when x \neq 3 and f(3)=k then the value of k is:

(a) 16 \log{2} (b) 4 \log{2} (c) 8\log{2} (d) 2 \log{2}

Problem 9:

A function f(x) is defined as follows:

f(x) = \frac{ab^{x}-ba^{x}}{x^{2}-1} where x \neq 1 and f(1)=k is continuous at x=1, then find the value of k.

Problem 10:

At the point x=0 the function f(x) where

f(x) = \frac{\log{\sec^{2}{(x)}}}{x \sin{x}}, when x \neq 0

f(x) =e when x=0 possesses

(a) removable discontinuity

(b) irremovable discontinuity

(c) no discontinuity

(d) none

Regards,

Nalin Pithwa

Limits and Continuity: Part 9: Tutorial Practice for IITJEE Maths

Problem 1:

If f(x) = \frac{(5^{\sin{x}}-1)^{2}}{x \log{(1+2x)}}, where x \neq 0 is continuous at zero, then find the value of f(0).

Problem 2:

If f(x) = 2x + a for 0 \leq x <1 and f(x) = 3x+b for 1 \leq x \leq 2 is continuous at x=1 and a+b=1, then the find the value of 3a-4b.

Problem 3:

If f(x) = \frac{2^{3x}-3^{x}}{x} for x<0 and f(x) = \frac{1}{x} \log{(1+ \frac{8x}{3})} for x>0.

Consider the following statements:

i) \lim_{x \rightarrow 0} f(x) does not exist.

ii) \lim_{x \rightarrow 0^{+}} f(x) exists but f(0) is not defined.

iii) f(x) is discontinuous at zero

iv) \lim_{x \rightarrow 0^{-}} f(x) exists, but f(0) is not defined.

Which of the above statements are false?

(a) all four (b) (ii) and (iv) (c) (i) and (iii) (d) none

Problem 4:

If f(x) = \frac{\log{x} - \log{2}}{x-2} for x >2 and f(x) = \frac{1-\cos{(x-2)}}{(x-2)^{2}} for x <2

Consider the following statements:

(i) \lim_{x \rightarrow 2^{-}} f(x) does not exist.

(ii) \lim_{x \rightarrow 2^{+}} does not exist.

(iii) f(x) is continuous at x=2

(iv) f(x) is discontinuous at x=2.

Which of the above statements are true?

(a) none (b) iv (c) iii (d) ii

Problem 5:

If the function f is continuous at x=0 and is defined by

f(x) = \frac{\sin{4x}}{5x}+a for x>0

f(x) = x+4-b for x <0

f(x) = 1 for x =0

The quadratic equation whose roots are values of 5a and 2b is

(a) x^{2}-2x+3=0 (b) x^{2} + 3x +2=0

(c) x^{2}-3x =2=0 (d) none

Problem 6:

The function f(x) = \frac{(e^{3x}-1)^{2}}{x \log{(1+3x)}} for x \neq 0 and f(0)=\frac{1}{3}

(a) has a removable discontinuity at x=0

(b) has irremovable discontinuity at x=0

(c) is continuous at x=0

(d) none of the above.

Problem 7:

If f(x) is continuous in [0,8] and

f(x) = x^{2} + ax + b when 0 \leq x <2

f(x) = 3x+2 when 2 \leq x \leq 4

f(x) = 2ax + 5b when 4 < x \leq 8

Then, values of a and b are:

(a) 3,2 (b) 1, -2 (c) -3, 2 (d) 3,-2

Problem 8:

The value of a^{2} - b^{2} if f is continuous on [-\pi, \pi] where

f(x) = -2\sin{x} for -\pi \leq x \leq -\frac{\pi}{2}

f(x) = a \sin{x} + b for -\frac{\pi}{2} < x < \frac{\pi}{2}

f(x) = \cos{x} for \frac{\pi}{2} \leq x \leq \pi is

(a) 0 (b) 2 (c) \infty (d) indeterminate

Problem 9:

Given f(x) = \frac{x^{2}+3x+5}{x^{2}-7x+10}. Let A \equiv [-2,3] and B \equiv [6,10] then

(a) f(x) is continuouis in B but discontinuous in A

(b) f(x) is discontinuous in B but continuous in A

(c) f(x) is continuous in both A and B

(d) f(x) is discontinuous in both A and B

Problem 10:

The function f(x) = \frac{2x^{2}-x+7}{x^{2}-4x+5} is

(a) continuous for all real values of x

(b) discontinuous for all real values of x

(c) discontinuous at x=1 and x=5

(d) discontinuous at x=2 and x=3

Regards,

Nalin Pithwa

Basic Partial Differentiation Tutorial: IITJEE Mains

Do the following: Find the partial derivatives of the following functions:

a) f(x,y,z) = x^{y}

b) f(x,y,z) = z

c) f(x,y) = \sin{(x \sin{y})}

d) f(x,y,z) = \sin{(x \sin{(y \sin{z})})}

e) f(x,y,z) = x^{y^{z}}

f) f(x,y,z) = x^{(y+z)}

g) f(x,y) = \sin{(xy)}

h) f(x,y,z) = (x+y)^{z}

i) f(x,y) = (\sin{(xy)})^{\cos {3}}

These are baby steps required to learn the techniques of solving differential equations.

Regards,

Nalin Pithwa

Limits and Continuity: part 8: IITJEE Math: Tutorial Problems for Practice

Problem 1:

Evaluate: \lim_{x \rightarrow \frac{\pi}{6}} \frac{2-\sqrt{3}\cos{x}-\sin{x}}{(6x-\pi)^{2}}

Problem 2:

Evaluate: \lim_{x \rightarrow 1} \frac{1-x^{2}}{\sin{(x\pi)}}

Problem 3:

Evaluate: \lim_{x \rightarrow 1}\frac{\cot{(\frac{\pi}{2}x)}}{x-1}

Problem 4:

Evaluate: \lim_{x \rightarrow 0} (1+\frac{4x}{5})^{\frac{10}{x}}

Problem 5:

Evaluate: \lim_{x \rightarrow 0} (\frac{1+ax}{1+bx})^{\frac{1}{x}}

Problem 6:

Evaluate: \lim_{x \rightarrow 0} (\frac{5+x}{5-x})^{\frac{1}{x}}

Problem 7:

Evaluate: \lim_{x \rightarrow 0} (\frac{4-3x}{4+5x})^{\frac{1}{x}}

Problem 8:

Evaluate: \lim_{x \rightarrow \infty} (1+ \frac{4}{n})^{3n}

Problem 9:

Evaluate: \lim_{x \rightarrow 1} \frac{\log{(2-x)}}{\sqrt{(3+x)}-2}

Problem 10:

Evaluate: \lim_{x \rightarrow 0} \frac{a^{x}-b^{x}}{3\sin{x} - \sin{(5x)}}

Problem 11:

Evaluate: \lim_{x \rightarrow 0} \frac{x \tan{x}}{e^{x}+e^{-x}-2}

Problem 12:

Evaluate: \lim_{x \rightarrow \frac{\pi}{4}} \frac{e^{(x - \frac{\pi}{4})}-1}{\cos{x} - \sin{x}}

Problem 13:

Evaluate: \lim_{x \rightarrow \frac{\pi}{2}} \frac{3^{(x - \frac{\pi}{2})} - 6^{(x - \frac{\pi}{2})}}{\cos{x}}

Problem 14:

Evaluate: \lim_{x \rightarrow 0} \frac{a^{3x}-a^{2x}-a^{x}+1}{x^{2}}

Problem 15:

Evaluate: \lim_{x \rightarrow 0} \frac{a^{x}+b^{x}-2^{(x+1)}}{x}

Problem 16:

Evaluate: \lim_{x \rightarrow \frac{\pi}{2}} \frac{2^{-\cos{x}}-1}{x(x - \frac{\pi}{2})}

Problem 17:

Evaluate: \lim_{\theta \rightarrow 0} \frac{3-4\cos{\theta}+\cos{2\theta}}{\theta^{4}}

Problem 18:

Evaluate: \lim_{x \rightarrow a} \frac{x \sin{a} - a \sin{x}}{x-a}

Problem 19:

Evaluate: \lim_{x \rightarrow 0} \frac{(27)^{x}-9^{x}-3^{x}+1}{\sqrt{2} - \sqrt{(1+\cos{x})}}

Problem 20:

Evaluate: \lim_{x \rightarrow 0} \frac{(5^{x}-2^{x})x}{\cos{5x} - \cos{3x}}

Problem 21:

Evaluate: \lim_{x \rightarrow 0} \frac{(3^{x}-1)^{2}}{2(1-\cos{x}) \log{(2+x)}}

Problem 22:

Evaluate: \lim_{x \rightarrow 1} \frac{\cos{(x \pi)} + \sin {(\frac{\pi}{2})x}}{(x-1)^{2}}

Problem 23:

Evaluate: \lim_{\theta \rightarrow \frac{\pi}{2}} \frac{\sin {\theta} + \cos{2 \theta}}{(\pi - 2 \theta)^{2}}

Problem 24:

Evaluate: \lim_{x \rightarrow \frac{1}{2}} \frac{2x^{2}+x-1}{4x^{2}-1+\sin{(2x-1)}}

Problem 25:

Evaluate: \lim_{x \rightarrow \infty} (\frac{2x+1}{2x-1})^{x+4}

Problem 26:

Evaluate: \lim_{x \rightarrow 0} \frac{e^{x} -2\cos{x} + e^{-x}}{x \sin{x}}

Problem 27:

Evaluate: \lim_{x \rightarrow 0} \frac{x^{2}}{\tan{x}} \sin{(\frac{1}{x})}

Problem 28:

Evaluate: \lim_{x \rightarrow 4} \frac{(\cos{\alpha})^{x} - (\sin{\alpha})^{x} -\cos{2\alpha}}{x-4}

There is one of the four possible answers:

(i) \log {(\frac{(\cos{\alpha})^{\cos^{-4}(\alpha)}}{(\sin{\alpha})^{\sin^{4}{(\alpha)}}})}

(ii) \log{(\frac{(\cos{\alpha})^{\cos^{4}{(\alpha)}}}{ (\sin{(\alpha)})^{\sin^{4}{(\alpha)}}})}

(iii) \log{(\frac{(\sin{\alpha})^{\sin^{4}{(\alpha)}}}{(\cos{(\alpha)})^{\cos^{4}{(\alpha)}}})}

(iv) \log{(\frac{(\sin{\alpha})^{\sin^{4}{\alpha}}}{(\cos{\alpha})^{\cos^{-4}{\alpha}}})}

Problem 29:

The values of A and B for f(x) to be continuous at x=0 where

f(x) = \frac{10^{x}+7^{x}-14^{x}-5^{x}}{1-\cos{x}} when x \neq 0

f(x) = \log{A} . \log{B} when x=0 are

(i) \frac{20}{7}, 1 (ii) \frac{10}{7}, 2 (iii) \frac{13}{7}, 1 (iv) \frac{5}{7}, 4

Problem 30:

If f(x) = \frac{\sqrt{1+\cos{x}}-1}{(\pi - x)^{2}} when x \neq \pi

and f(x) = k when x=\pi

Find the value of k for which f(x) is continuous at \pi.

Regards,

Nalin Pithwa

Limits and Continuity: part 7: IITJEE math: Tutorial Problems for Practice

Problem 1:

Find the value of the following limit: \lim_{\theta \rightarrow \frac{\pi}{4}} \frac{2- cosec(\theta)*cosec(\theta)}{1-\cos{\theta}}

Problem 2:

Find the value of the following limit: \lim_{x \rightarrow 2}\frac{2x^{2}-7x+6}{5x^{2}-11x+2}

Problem 3:

Find the value of the following limit: \lim_{x \rightarrow 4} \frac{x^{4}-64x}{\sqrt{(x^{2}+9)}-5}

Problem 4:

Find the value of the following limit: \lim_{x \rightarrow 2} (\frac{1}{x-2} + \frac{6x}{8-x^{3}})

Problem 5:

Find the value of the following limit: \lim_{x \rightarrow \infty} \frac{4x^{4}-3x^{3}+2x^{2}-x+1}{3x^{4}-2x^{3}+x^{2}-x-7}

Problem 6:

Find the value of the following limit: \lim_{x \rightarrow \infty}(\sqrt{x^{2}+4x+5} -\sqrt{x^{2}+1})

Problem 7:

Find the following limit: \lim_{h \rightarrow 0} \frac{f(1+h)-f(1)}{h} where f(x) = \sqrt{7-2x}

Problem 8:

Evaluate: \lim_{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots+x^{2n} -2n}{x-1}, where n \in N

Problem 9:

Evaluate: \lim_{x \rightarrow 0} \frac{1-\cos{(2x)}}{\cos{(2x)}-\cos{(8x)}}

Problem 10:

Evaluate: \lim_{\theta \rightarrow 0} \frac{5\theta\cos{\theta}-2\sin{\theta}}{3\theta+\tan{\theta}}

Problem 11:

Evaluate: \lim_{x \rightarrow 0} \frac{3\sin {(x \deg)}- \sin{(3x \deg)}}{x^{3}}

Problem 12:

Evaluate: \lim_{x \rightarrow 0} \frac{1-\cos{(\frac{x}{2})}}{1-\cos{(\frac{x}{3})}}

Problem 13:

Evaluate: \lim_{x \rightarrow 0} \frac{\cos{x} - \sqrt{(\cos{x})}}{x^{2}}

Problem 14:

Evaluate: \lim_{x \rightarrow 0} \frac{5\sin{x}-7\sin{2x}+3\sin{3x}}{x^{2}\sin{x}}

Problem 15:

Evaluate: \lim_{x \rightarrow 0} \frac{x^{2}+1-\cos{x}}{x\tan{x}}

Problem 16:

Evaluate: \lim_{x \rightarrow \frac{\pi}{6}} \frac{\cos{x} - \sqrt{3}\sin{x}}{\pi - 6x}

Problem 17:

Evaluate: \lim_{x \rightarrow a} \frac{\sin{(\sqrt{x})}-sin{(\sqrt{a})}}{x-a}

Problem 18:

Evaluate: \lim_{x \rightarrow 1} \frac{1+ \cos{(x\pi)}}{(1-x)^{2}}

Problem 19:

Evaluate: \lim_{x \rightarrow 0}(1+\sin{x})^{\frac{1}{x}}

Problem 20:

Evaluate: \lim_{x \rightarrow 0}(\frac{3+2x}{3-x})^{\frac{1}{x}}

Problem 21:

Evaluate: \lim_{x \rightarrow 1} x^{\frac{1}{x-1}}

Problem 22:

Evaluate: \lim_{x \rightarrow 0} (1+x+\frac{x^{2}}{4})^{\frac{1}{x}}

Problem 23:

Evaluate: \lim_{x \rightarrow 0} (\tan{(\frac{\pi}{4}+x)})^{\frac{1}{x}}

Problem 24:

Evaluate: \lim_{x \rightarrow 0} \frac{\log{(e^{2}+x^{2})}-2}{1-\cos{(2x)}}

Problem 25:

Evaluate: \lim_{x \rightarrow 0} \frac{5^{x}-3^{x}}{4^{x}-1}

Problem 26:

Evaluate: \lim_{x \rightarrow 0} \frac{12^{x}-4^{x}-3^{x}+1}{x \tan{x}}

Problem 27:

Evaluate: \lim_{x \rightarrow 0} \frac{3^{x}+3^{-x}-2}{(2^{x}-1)(\log{(1+x)})}

Problem 28:

Evaluate: \lim_{x \rightarrow 0} \frac{(3^{x}-2^{x})^{2}}{1-\cos{(2x)}}

Problem 29:

Evaluate: \lim_{x \rightarrow 0} \frac{a^{x}+b^{x}+c^{x}-3^{(x+1)}}{\sin{x}}

Problem 30:

Evaluate : \lim_{x \rightarrow 0} \frac{(3^{x}-1)^{3}}{(2^{x}-1)(\sin{x})(\log{(1+x)})}

Problem 31:

Evaluate: \lim_{x \rightarrow 1} \frac{4^{x-1}-2^{x}+1}{(x-1)^{2}}

Problem 32:

Evaluate: \lim_{x \rightarrow 2} \frac{4^{x-2}-2^{x-1}+1}{(x-2)(\log{(x-1)})}

Problem 33:

Evaluate: \lim_{x \rightarrow 0} \frac{9^{x}-2 \times 3^{x}+1}{1-\cos{x}}

Problem 34:

Evaluate: \lim_{x \rightarrow 0} \frac{10^{x}+7^{x}-14^{x}-5^{x}}{x^{2}}

Problem 35:

Evaluate: \lim_{x \rightarrow 0} \frac{(2^{\sin{x}}-1)^{2}}{x \log{(1-x)}}

Problem 36:

Evaluate: \lim_{x \rightarrow 1} \frac{ab^{x}-ba^{x}}{(x-1)}

Problem 37:

Evaluate: \lim_{x \rightarrow 2} \frac{ax^{2}-b}{x-2} = 4. Then, (i) a=1, b=4 (ii) a=4, b=1 (iii) a=-4, b=1 (iv) a=2, b=1

Problem 38:

Evaluate: \lim_{x \rightarrow 2}\frac{x^{4}-8x}{\sqrt{x^{2}+21}-5}

Problem 39:

Evaluate: \lim_{x \rightarrow 2a} \frac{\sqrt{x-2a}+\sqrt{x} -\sqrt{2a}}{\sqrt{x^{2}-4a^{2}}}

Problem 40:

Evaluate: \lim_{x \rightarrow 4} \frac{x^{3}-64}{x^{3}-15x-4}

Problem 41:

Evaluate: \lim_{x \rightarrow 3} \frac{x^{2}+\sqrt{x+6}-12}{x^{2}-9}

Problem 42:

Evaluate: \lim_{x \rightarrow 2} \frac{x^{3}+\sqrt{x+2}-10}{x^{2}-4}

Problem 43:

Evaluate: \lim_{x \rightarrow 2} (\frac{1}{x-2} - \frac{2}{x^{3}-3x^{2}+2x})

Problem 44:

Evaluate: \lim_{x \rightarrow \infty} \sqrt{x} (\sqrt{x+2}-\sqrt{x})

Problem 45:

Evaluate: \lim_{h \rightarrow 0} \frac{h}{(a+h)^{8}-a^{8}}

Problem 46:

Evaluate: \lim_{x \rightarrow 1} \frac{x^{4}+x^{7}-2}{x^{3}-2x+1}

Problem 47:

Evaluate: \lim_{x \rightarrow 3} \frac{x+x^{2}+x^{3}-39}{x-3}

Problem 48:

Evaluate: \lim_{x \rightarrow \frac{\pi}{4}} \frac{2- cosec (x) * cosec(x)}{\cot{x}-1}

Problem 49:

Evaluate: \lim_{x \rightarrow 1} \frac{(x^{2}+x) \sin{(x-1)}}{x^{2}+x-2}

Problem 50:

Evaluate: \lim_{x \rightarrow 0} \frac{\cos{8x} - \cos{2x}}{\cos{12x}-\cos{4x}}

Regards,

Nalin Pithwa

Limits and continuity: part 6: IITJEE math: Tutorial Problems for Practice

Problem 1:

Find the value of the following limit:

\lim_{y \rightarrow x} \frac{y^{y}-x^{x}}{y-x}

Problem 2:

Find the value of the following limit:

\lim_{x \rightarrow 2} \frac{e^{\log {(\frac{3x+4}{x})}} - e^{\log{5}}}{x-2}

Problem 3:

Find the value of the following limit:

\lim_{h \rightarrow 0} \frac{(a+h)^{2} \sin{(a+h)}-a^{2}\sin{a}}{h}. Choose one of the following: (i) a\cos{a} -2 \sin{a} (ii) a \cos{a} +2\sin{a} (iii) a(a\cos{a} + 2 \sin{a}) (iv) a^{2} \cos{a}

Problem 4:

Find the value of the following limit:

\lim_{x \rightarrow -2} \frac{x^{5}+2x^{4}+x^{2}+3x+2}{x^{4}+2x^{3}+3x^{2}-5x-22}

Problem 5:

Find the value of the following limit:

\lim_{x \rightarrow \infty} (\frac{x+1}{x+2})(\frac{2x+1}{3x+4})

Problem 6:

Find the value of the following limit:

\lim_{x \rightarrow \infty} \frac{(x+1)^{10}+(x+2)^{10} + \ldots + (x+100)^{10}}{(x^{10}+10^{10})}

Problem 7:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{\cos{(ax)}-\cos{(bx)}}{\cos{(cx)} - \cos{(dx)}}

Problem 8:

Find the value of the following limit:

\lim_{x \rightarrow \pi} \frac{1-\cos{(7(x-\pi))}}{5(x-\pi)^{2}}

Problem 9:

Find the value of the following limit:

\lim_{x \rightarrow 1} \frac{\sqrt[3]{x^{2})-2.\sqrt[3]{(x)}+1}()}{(x-1)^{2}}

Problem 10:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{\sec{(4x)} - \sec{(2x)}}{(\sec{(3x)}-\sec{x})}

Problem 11:

Find the value of the following limit:

\lim_{x \rightarrow 1} \frac{1+\cos{\pi x}}{\tan^{2}{\pi x}}

Problem 12:

Find the value of the following limit:

\lim_{x \rightarrow 1} \frac{3 \sin{x\pi } -\sin{3x \pi}}{(x-1)^{3}}

Problem 13:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{\tan^{4}{x} - \sin^{4}{x}}{x^{6}}

Problem 14:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{x^{3}\sin{x}}{(\sec{x} - \cos{x})^{2}}

Problem 15:

If the value of the following limit is -1, then find the value of a:

\lim_{x \rightarrow a} \frac{\sin{x} -\sin{a}}{\cos{x} -\cos{a}}

Problem 16:

Find the value of the following limit:

\lim_{x \rightarrow 0} (\tan{(\frac{\pi}{4}-x)})^{\frac{1}{x}}

Problem 17:

Find \lim_{x \rightarrow 0} \frac{f^{'}(x)}{x} if f(x) is given as follows:

f(x) = \left | \begin{array}{ccc} \cos{x} & x & 1 \\ 2\sin{x} & x^{2} & 2x \\ \tan{x} & x & 1 \end{array} \right |

Problem 18:

If f(a) = \lim_{x \rightarrow \infty} x(a^{\frac{1}{x}}-1), then f(ab) is equal to (i) f(a).f(b) (ii) f(a)+f(b) (iii) 0 (iv) ab

Problem 19:

Evaluate the following limit:

\lim_{x \rightarrow 1} (\frac{x^{3}+2x^{2}+x+1}{x^{2}=2x+3})^{\frac{1-\cos{(x-1)}}{(x-1)^{2}}}

Problem 20:

The function f is defined by :

f(x) = \frac{e^{x}+e^{-x}-2}{x \sin{x}} in the interval [\frac{\pi}{2}, - \frac{\pi}{2}]-\{0\}

In order for this function to be continuous in [\frac{\pi}{2}, -\frac{\pi}{2}], we have to define (a) f(0)=2 (b) f(\frac{\pi}{2})=1 (c) f(0)=1 (d) f(-\frac{\pi}{2})=1

Problem 21:

The function f(x) = \frac{|x|}{x} when x \neq 0 and f(0)=0, (a) has removable discontinuity at x=0 (b) has irremovable discontinuity at x=0 (c) is continuous at x=0 (d) \lim_{x \rightarrow 0}f(x) exists.

Problem 22:

Let f(x) be defined by

f(x) = \sin{2x} if 0 < x \leq \frac{\pi}{6}

f(x) = ax + b, if \frac{\pi}{6} < x \leq 1

If f(x) and f^{'}(x) are continuous in (0,1), then the value of b is (i) \frac{1}{2} - \frac{\pi}{6} (ii) \frac{1}{2} - \frac{\pi}{3} (iii) \frac{\sqrt{3}}{2} + \frac{\pi}{6} (iv) \frac{\sqrt{3}}{2} - \frac{\pi}{6}

Problem 23:

If f(x) = x^{\frac{2}{3}}-2, where x \geq 0, then \lim_{x \rightarrow 2} f^{-1}(x) is (a) 7 (b) 8 (c) 9 (d) 10

Problem 24:

If f(x) = \frac{x+1}{x+2} and g(x) = \frac{1}{x} then \lim_{x \rightarrow 2} (f+g)(x) is (a) \frac{1}{5} (b) \frac{2}{5} (c) \frac{3}{5} (d) \frac{4}{5}

Regards,

Nalin Pithwa

Limits and Continuity: part 5: IITJEE Math: Tutorial problems for practice

Problem 1:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{\sin{a} - \tan{a}}{\sin^{a}}

Problem 2:

Find the values of the constant a and b such that the following limit is zero:

\lim_{x \rightarrow \infty} [\frac{x^{2}+1}{x+1} -ax-b]

Problem 3:

Find the value of the following limit:

\lim_{\alpha \rightarrow \beta} \frac{\sin^{2}{\alpha}-\sin^{2}{\beta}}{\alpha^{2}-\beta^{2}}

Problem 4:

If a, b, c, d are positive, then find the value of the following limit:

\lim_{x \rightarrow \infty}(1+\frac{1}{a+bx})^{c+dx}

Problem 5:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{(1-\cos{(2x)})\sin{(5x)}}{x^{2}\sin{(3x)}}

Problem 6:

Find the value of the following limit:

\lim_{x \rightarrow \infty} \frac{\sqrt{x^{2}-1}}{2x+1}

Problem 7:

Find the value of the following limit:

\frac{\log{(1+x+x^{2})}+\log{(1-x+x^{2})}}{\sec{x}-\cos{x}}

Problem 8:

Find the value of the following limit:

\lim_{x \rightarrow \infty} (\frac{2+x}{1+x})^{2x+1}

Problem 9:

Find the value of f(0) such that the following function is continuous at zero:

f(x) = (x+1)^{\cot{x}}

Problem 10:

Let f^{''}(x) be continuous at zero and f^{''}(0)=4. Then, find the numerical value of the following limit:

\lim_{x \rightarrow 0}\frac{2f(x)-3f(2x)+f(4x)}{x^{2}}

Problem 11:

Find the value of the following limit:

\lim_{n \rightarrow \infty} (\frac{n^{3}}{3n^{2}-4} - \frac{n^{2}}{3n+2})

Problem 12:

Find the values of x where the following function is discontinuous:

f(x) = \frac{\sin{x} \log{(x-2)}}{(x^{2}-4x+3)}

Problem 13:

The value of p for which the following function may be continuous at zero is what:

f(x) = \frac{(4x-1)^{3}}{(\sin{\frac{x}{p}})(\log{(1+\frac{x^{2}}{3})})}, when x \neq 0, and

f(x) = 12(\log{4})^{3}, when x = 0.

Problem 14:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{1-\cos{(mx)}}{1-\cos{(nx)}}

Problem 15:

If f(x) = \frac{4-7x}{3x+4} and \lim_{x \rightarrow 2}f(x) = k, and \lim_{x \rightarrow 0}f(x) = m, then the equation whose roots are \frac{1}{k}, \frac{1}{m} is (a) x^{2}+x=0 (b) x^{2}-1=0 (c) x^{2}+1=0 (d) x^{2}+2x=0

Problem 16:

Find the value of the following limit:

\lim_{x \rightarrow 1} \frac{x+x^{2}+x^{3}+\ldots + x^{n}-n}{x-1}

Problem 17:

Find the value of the following limit:

\lim_{x \rightarrow 1} \frac{\sqrt[n]{x^{m}}-1}{\sqrt[m]{x^{n}}-1}

Problem 18:

Find the value of the following limit:

\lim_{x \rightarrow a} \frac{\tan{x} - \tan{a}}{\sin{a} - \sin{x}}

Regards,

Nalin Pithwa

Limits and Continuity: IITJEE Maths : Tutorial problems 1

Problem 1: Which of the following is an indeterminate form ? (a) 1^{1} (b) 0^{1} (c) 1^{0} (d) 0^{0}

Problem 2: Which of the following is not an indeterminate form ? (a) 1^{1} (b) 0 \times \infty (c) 1^{\infty} (d) \infty^{0}

Problem 3: If \lim_{x \rightarrow c} f(x) and \lim_{x \rightarrow c}g(x) exists then which of the following conditions is not always correct ? (i) \lim_{x \rightarrow c}(f(x)+g(x)) = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c}g(x) (ii) \lim_{x \rightarrow c}(f(x)-g(x)) = \lim_{x \rightarrow c}f(x) - \lim_{x \rightarrow c}g(x) (iii) \lim_{x \rightarrow c}(f(x)g(x)) = \lim_{x \rightarrow c}f(x) \times \lim_{x \rightarrow c}g(x) (iv) \lim_{x \rightarrow c} (\frac{f(x)}{g(x)}) = \frac{\lim_{x \rightarrow c}f(x)}{\lim_{x \rightarrow c}g(x)}

Problem 4: If \lim_{x \rightarrow c}(\frac{f(x)}{g(x)}) exists, then (i) both \lim_{x \rightarrow c}f(x) and \lim_{x \rightarrow a}g(x) must exist (ii) \lim_{x \rightarrow a}f(x) need not exist but \lim_{x \rightarrow a}g(x) exists. (iii) neither \lim_{x \rightarrow a}f(x) nor \lim_{x \rightarrow a}g(x) may exist (d) \lim_{x \rightarrow a}f(x) exists but \lim_{x \rightarrow a}g(x) need not exist.

Problem 5: \lim_{x \rightarrow a+}f(x)=l=\lim_{x \rightarrow a-}g(x) and \lim_{x \rightarrow a-}f(x) = m = \lim_{x \rightarrow a+}g(x) then the function (f(x)-g(x)) (i) is continuous at x=a (ii) is not continuous at x=a (iii) has a limit when x \rightarrow a but \lim_{x \rightarrow a}(f(x)-g(x))=l-m (iv) has a limit equal to l-m when x \rightarrow a

Problem 6: If \lim_{x \rightarrow a+}f(x) = l = \lim_{x \rightarrow a-}g(x) and \lim_{x \rightarrow a-}f(x)=m=\lim_{x \rightarrow a+}g(x) then the function (f(x).g(x)) (i) is continuous at x=a (ii) does not have a limit at x=a (iii) has a limit when x \rightarrow a and it is equal to l.m (iv) has a limit when x \rightarrow a but it is not equal to l.m

Problem 7: Find \lim_{x \rightarrow \frac{3.\pi}{4}}\frac{1+\tan{x}}{\cos{(2x)}}

Problem 8: Find \lim_{x \rightarrow e}\frac{\log{x}-1}{x-e}.

Problem 9: Find \lim_{x \rightarrow 0}\frac{a^{x}-b^{x}}{x}

Problem 10: Find \lim_{x \rightarrow 0}\frac{2(1-\cos{x})}{x^{2}}.

Problem 11: Find \lim_{x \rightarrow 0}\frac{\sqrt{1+x}-1}{x}

Problem 12: Find \lim_{x \rightarrow 3-}\frac{|x-3|}{x-3}

Problem 13: Find \lim_{x \rightarrow 0}(\frac{1+\tan{x}}{1+\sin{x}})^{cosec{x}}

Problem 14: Find \lim_{x \rightarrow 0} \frac{(e^{2\sqrt{x}}-1)(\tan{3\sqrt{x}})}{\sin{x}}

Problem 15: Find \lim_{x \rightarrow 0}\frac{\log{\cos{x}}}{x}

Problem 16: Find x if \lim_{x \rightarrow a}\frac{a^{x}-x^{a}}{x^{x}-a^{a}}=-1

Regards,

Nalin Pithwa

best explanation of epsilon delta definition

Refer any edition of (i) Calculus and Analytic Geometry by Thomas and Finney (ii) recent editions which go by the title “Thomas’ Calculus”. If you need, you will have to go through the previous stuff (given in the text) on “preliminaries” and/or functions also. For Sets, Functions and Relations, I have also presented a long series of articles on this blog.

Ref:

 

https://www.amazon.in/Thomas-Calculus-George-B/dp/9353060419/ref=sr_1_1?crid=3F1XO0L9KBT1F&keywords=thomas+calculus&qid=1581323971&s=books&sprefix=Thomas+%2Caps%2C265&sr=1-1

Set theory, relations, functions: preliminaries: Part V

Types of functions: (please plot as many functions as possible from the list below; as suggested in an earlier blog, please use a TI graphing calculator or GeoGebra freeware graphing software): 

  1. Constant function: A function f:\Re \longrightarrow \Re given by f(x)=k, where k \in \Re is a constant. It is a horizontal line on the XY-plane.
  2. Identity function: A function f: \Re \longrightarrow \Re given by f(x)=x. It maps a real value x back to itself. It is a straight line passing through origin at an angle 45 degrees to the positive X axis.
  3. One-one or injective function: If different inputs give rise to different outputs, the function is said to be injective or one-one. That is, if f: A \longrightarrow B, where set A is domain and set B is co-domain, if further, x_{1}, x_{2} \in A such that x_{1} \neq x_{2}, then it follows that f(x_{1}) \neq f(x_{2}). Sometimes, to prove that a function is injective, we can prove the conrapositive statement of the definition also; that is, y_{1}=y_{2} where y_{1}, y_{2} \in codomain \hspace{0.1in} range, then it follows that x_{1}=x_{2}. It might be easier to prove the contrapositive. It would be illuminating to construct your own pictorial examples of such a function. 
  4. Onto or surjective: If a function is given by f: X \longrightarrow Y such that f(X)=Y, that is, the images of all the elements of the domain is full of set Y. In other words, in such a case, the range is equal to co-domain. it would be illuminating to construct your own pictorial examples of  such a function.
  5. Bijective function or one-one onto correspondence: A function which is both one-one and onto is called a bijective function. (It is both injective and surjective). Only a bijective function will have a well-defined inverse function. Think why! This is the reason why inverse circular functions (that is, inverse trigonometric functions have their domains restricted to so-called principal values). 
  6. Polynomial function: A function of the form f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\ldots + a_{n}x^{n}, where n is zero or positive integer only and a_{i} \in \Re is called a polynomial with real coefficients. Example. f(x)=ax^{2}=bx+c, where a \neq 0, a, b, c \in \Re is called a quadratic function in x. (this is a general parabola).
  7. Rational function: The function of the type \frac{f(x)}{g(x)}, where g(x) \neq 0, where f(x) and g(x) are polynomial functions of x, defined in a domain, is called a rational function. Such a function can have asymptotes, a term we define later. Example, y=f(x)=\frac{1}{x}, which is a hyperbola with asymptotes X and Y axes.
  8. Absolute value function: Let f: \Re \longrightarrow \Re be given by f(x)=|x|=x when x \geq 0 and f(x)=-x, when x<0 for any x \in \Re. Note that |x|=\sqrt{x^{2}} since the radical sign indicates positive root of a quantity by convention.
  9. Signum function: Let f: \Re \longrightarrow \Re where f(x)=1, when x>0 and f(x)=0 when x=0 and f(x)=-1 when x<0. Such a function is called the signum function. (If you can, discuss the continuity and differentiability of the signum function). Clearly, the domain of this function  is full \Re whereas the range is \{ -1,0,1\}.
  10. In part III of the blog series, we have already defined the floor function and the ceiling function. Further properties of these functions are summarized (and some with proofs in the following wikipedia links): (once again, if you can, discuss the continuity and differentiablity of the floor and ceiling functions): https://en.wikipedia.org/wiki/Floor_and_ceiling_functions
  11. Exponential function: A function f: \Re \longrightarrow \Re^{+} given by f(x)=a^{x} where a>0 is called an exponential function. An exponential function is bijective and its inverse is the natural logarithmic function. (the logarithmic function is difficult to define, though; we will consider the details later). PS: Quiz: Which function has a faster growth rate — exponential or a power function ? Consider various parameters.
  12. Logarithmic function: Let a be a positive real number with a \neq 1. If a^{y}=x, where x \in \Re, then y is called the logarithm of x with base a and we write it as y=\ln{x}. (By the way, the logarithmic function is used in the very much loved mp3 music :-))

Regards,

Nalin Pithwa