Limits and Continuity: part 11: IITJEE Maths tutorial problems for practice

Problem 1:

A function f(x) is defined as follows:

f(x) = \frac{(e^{2x}-1)(1-\cos{x})}{\tan^{2}{(x)}\log{(1+2x)}} when x \neq 0

f(0) = \log{a} is continuous at x=0.

The value of a should be

(i) \frac{e}{2} (b) \frac{1}{2e} (c) 2 (d) none

Problem 2:

If f(x) = \frac{e^{(2x)} + e^{(-2x)} -2}{1-\cos{(4x)}} when x \neq 0 is continuous at x=0, then what is the value of f(0)

Problem 3:

Given f(x) = x + a, when -1  \leq x \leq 0

and f(x) = x + b when 0 < x \leq 1

and f(x) = c -x when 1 < x \leq 2

if f is continuous at x=0 and x=1 and f(2)=1, then the value of 3a+b-2c=

(i) 0 (ii) 1 (iii) 2 (iv) 3

Problem 4:

If the function f(x) is continuous on its domain where

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

f(x)=4x-1 for 2 \leq x < 4

f(x)=ax^{2+17b} for 4 \leq x \leq 6

then the quadratic equation whose roots are 2a and 2b is:

(i) x^{2}+2x-8 (b) x^{2}-2x-8=0 (c) x^{2}+2x+8 (d) x^{2}-2x+8=0

Problem 5:

The value of c for which the function

f(x) = \frac{\sin{(x)} + \sin{((a+1)x)}}{x} when x<0

f(x) = c when x=0

f(x) = \frac{(x+bx^{2})^{\frac{1}{2}}-x^{\frac{1}{2}}}{bx^{\frac{3}{2}}}

is continuous at x=0 is

(i) 1/2 (ii) -1/2 (iii) 2 (iv) -2

Problem 6:

If f(x) = \frac{\sin{x\pi}}{x-1}+a when x<1

f(x) = 2x, when x=1

f(x)= \frac{1+\cos{x\pi}}{\pi (1-x)^{2}} + b when x>1

is continuous at x=1, then a and b have the values:

(i) 3\pi, 3\frac{\pi}{2} (ii) 3\pi, \frac{\pi}{2} (iii) \pi, \frac{\pi}{2} (iv) \pi, 3\frac{\pi}{2}

Problem 7:

If f(x) = \frac{(\sin{x} - \cos{x})^{2}}{\sqrt{2}-\sin{x}-\cos{x}}, when x \neq \frac{\pi}{4} is continuous at x=\frac{\pi}{4} then f(\frac{\pi}{4})=

(a) 1/2 (b) -1/2 (c) 2 (d) none of these

Problem 8:

If f(x)= \frac{x+1}{x+2} and g(x)=\frac{1}{x}, then \lim_{x \rightarrow 2} (g+f)(x)=

(i) 4/3 (b) 5/3 (c) 2 (d) 7/3

Problem 9:

Evaluate the following: \lim_{x \rightarrow 4} \frac{(x^{2}-x-12)^{18}}{(x^{3}-8x^{2}+16x)^{9}}

Regards,

Nalin Pithwa

Wisdom of George Polya

A great discovery solves a great problem. But there is a grain of discovery in the solution of any problem.

— George Polya, How to Solve it

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: Part 4: IITJEE Math Tutorial Problems for Practice

Problem 1:

If \alpha, \beta are the two roots of the quadratic equation ax^{2}+bx+c=0, then the find the value of the following limit:

\lim_{x \rightarrow \alpha} \frac{1-\cos{(ax^{2}+bx+c)}}{(x-\alpha^{2})}

Problem 2: Given the following functio; find the value of f(0) so that the function is continuous at zero:

f(x) = \frac{\sqrt{1+x}-(1+x)^{\frac{1}{3}}}{x} when x \neq 0.

Problem 3:

Find the value of the following limit: \lim_{x \rightarrow 0} \frac{\sin {(x \deg)}}{x}

Problem 4:

If \lim_{x \rightarrow 0}\frac{1-\cos{(1-\cos{x})}}{x^{4}}=k, which numerical value divides k^{-2}?

Problem 5:

Find the value of the following limit:

\lim_{x \rightarrow 1} {(\sec{(\frac{\pi x}{2})})(\log{x})}

Problem 6:

Find the value of the following limit:

\lim_{x \rightarrow \infty} \frac{(2+x)^{40}(4+x)^{5}}{(2-x)^{45}}

Problem 7:

Let it be given that L = \lim_{x \rightarrow 2} (x^{3}-x^{2}+x-1) and M = \lim_{x \rightarrow -2}(x^{4}-x^{3}+x^{2}-x) then find the value of the following limit:

\lim_{x \rightarrow 1}\frac{Lx^{2}-Mx+2}{Mx^{2}-Lx-2}

Problem 8:

Find the value of the following limit:

\lim_{x \rightarrow \arctan{-3}} \frac{\tan^{2}{x}-2\tan{x}-3}{\tan^{2}-4\tan{x}+3}

Problem 9:

Find the value of the following limit:

\lim_{x \rightarrow \infty} (\frac{x+6}{x+1})^{x+4}

Problem 10:

Find the value of b such that the following function is continuous at every point of its domain:

f(x) = 5x-4, when 0 < x \leq 1, and f(x)=4x^{2}+3bx, when 1 < x < 2.

Problem 11:

Find the value of the following limit:

\lim_{x \rightarrow 0}(\cos{x})^{\frac{1}{x}}

Problem 12:

Find the value of the following limit:

\lim_{h \rightarrow 0} \frac{\tan{(x+h)-\tan{x}}}{x}

Problem 13:

Consider the following: \lim_{x \rightarrow 0}\frac{\sqrt{1-\cos{2x}}}{x}. Then, which of the following is true (a) limit exists and is equal to \sqrt{2} (b) exists and it equals -\sqrt{2} (c) limit does not exist because x - 1 \rightarrow 0 (d) limit does not exist because left hand limit is not equal to right hand limit

Problem 14:

Find the value of the following limit:

\lim_{x \rightarrow \frac{\pi}{4}} (\frac{1-\tan{x}}{1-\sqrt{2}\sin{x}})

Problem 15:

Find the value of p given the following:

\lim_{x \rightarrow 0} \frac{\sin{px}}{\tan{3x}}=4

Problem 16:

The number of points of discontinuity of the function f(x)= \frac{1}{\log{|x|}} is (a) zero (b) 1 (c) 2 (d) 3

Problem 17:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{\sin(\pi \cos^{2}{x})}{x^{2}}

Problem 18:

Find the value of the following limit:

\lim_{x \rightarrow 2}\frac{3^{\frac{x}{2}}-3}{3^{x}-9}

Regards,

Nalin Pithwa