Basic Partial Differentiation Tutorial: IITJEE Mains

October 16, 2020 – 1:14 pm

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

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

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

October 10, 2020 – 12:34 pm

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

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

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

October 10, 2020 – 8:30 am

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

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

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

October 10, 2020 – 4:05 am

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

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

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

October 10, 2020 – 3:16 am

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

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)

Limits and Continuity: Part 4: IITJEE Math Tutorial Problems for Practice

October 10, 2020 – 2:43 am

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

By Nalin Pithwa | Posted in IITJEE Mains | Comments (0)

Limits and Continuity: Part 3: IITJEE maths tutorial problems

October 9, 2020 – 6:54 pm

Problem 1:

Find the following limit:

\lim_{h \rightarrow 0} 2 \times \frac{\sqrt{3}(\sin{(\frac{\pi}{6}+h)})-\cos{(\frac{\pi}{6}+h)}}{\sqrt{3}h(\sqrt{3}\cos{h}-\sin{h})}

Problem 2:

Let the given function be continuous in the interval [-1,1]. Then what must be the value of p?

f(x) = \frac{\sqrt{(1+px)}-\sqrt{(1-px)}}{x}, when -1 \leq x \leq 0

f(x) = \frac{2x+1}{x-2}, when 0 \leq x \leq 1.

Problem 3:

Let the given function be continuous for 0 \leq x < \infty, then find the most suitable values for a and b:

f(x) = \frac{x^{2}}{a}, for 0 \leq x <1

f(x) = a, for 1 \leq x < \sqrt{2}

f(x) = \frac{2b^{2}-4b}{x^{2}}, for \sqrt{2} \leq x < \infty

Problem 4:

Find the value of the following:

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

Problem 5:

The function f(x) = \frac{1}{x} \times (\sqrt{(1+\sin{x})} - \sqrt{(1-\sin{x})}) is not defined at x=0. The value of f(0) so that f(x) becomes continuous at x=0 is (a) 1 (b) 2 (c) 0 (d) none

Problem 6:

Find the value of the following limit:

\lim_{x \rightarrow 0} \frac{a^{x}-1}{\sqrt{(1+x)}-1}

Problem 7:

Let the given function be f(x) = \frac{\tan{(\frac{\pi}{4}-x)}}{\cot{(2x)}}. Find the value which should be assigned to f at x = \frac{\pi}{4} so that f is continuous everywhere on the reals.

Problem 8:

Let it be given that n \in N and f(x) = \frac{1+2^{x}+3^{x}+\ldots + n^{x}-n}{x}, x is not zero. What value of f(0) will make the function f continuous on the reals.

Problem 9:

Find the value of the following limit:

\lim_{\theta \rightarrow 0^{+}}\frac{\sin{\sqrt{\theta}}}{\sqrt{(\sin{\theta})}}

Problem 10:

If a = \log_{3}{(3x)} and b = \log_{x}{(3)}, then the find the limiting value of a^{b} as x \rightarrow 1:

Problem 11:

Let it be given that n \in N. Then, the find the value of the following limit:

\lim_{x \rightarrow 0}\frac{\sin{x}+\sin{(2x)}+\ldots + \sin{(nx)}}{\sin{x}+\sin{(3x)}+\sin{(5x)}+\ldots + \sin{(2n-1)x}}

Problem 12:

Let it be given that f(x) = x \sin{(\frac{1}{x})} when x is not zero and f(x) = 0, when x is zero. Then, find the value of the following limit:

\lim_{x \rightarrow 0}f(x).

Problem 13:

Find the value of the following limit:

\lim_{x \rightarrow 0}\frac{e^{x^{2}}-\cos{(x)}}{x^{2}}

Problem 14:

Let it be given that f(x) = \frac{x^{2}-(A+2)x+A}{x-2} when x \neq 2 and f(x) = 2, when x=2 is continuous at x=2. Then, find the value of A.

Regards,

Nalin Pithwa


By Nalin Pithwa | Posted in IITJEE Mains | Comments (0)

Why writing by hand makes kids smarter

October 2, 2020 – 3:08 am
https://medicalxpress.com/news/2020-10-kids-smarter.html#:~:text=Handwriting%20gives%20the%20brain%20more%20hooks%20to%20hang%20memories%20on&text=The%20results%20showed%20that%20the,to%20hang%20your%20memories%20on.
By Nalin Pithwa | Posted in memory power concentration retention, miscellaneous, motivational stuff, time management | Comments (0)

Limits and Continuity: IITJEE Maths: Tutorial Problem Set 2

September 27, 2020 – 6:48 am

Problem 1: Find \lim_{x \rightarrow \infty}(1+\frac{2}{x})^{x}.

Problem 2: If G(x)=-\sqrt{25-x^{2}}, then what is the value of \lim_{x \rightarrow 1} \frac{G(x)-G(1)}{x-1}?

Problem 3: If f(x) = (1-x)\tan{\frac{\pi}{2}}, then find the value of \lim_{x \rightarrow 1}f(x)

Problem 4: Find the value of \lim_{x \rightarrow 2}\frac{\sqrt{(x^{2}+5)} -3}{(x-2)}

Problem 5: Find the value of \lim_{x \rightarrow \frac{\pi}{4}}(\sin{2x})^{\tan^{2}{2x}}

Problem 6: Find the value of \lim_{x \rightarrow 0}\frac{\sin{x} - x}{x^{3}}

Problem 7: Find the value of \lim_{n \rightarrow \infty}

Problem 8: If m, n \in N, what is the relationship between m and n if \lim_{x \rightarrow 0} \frac{(\sin)^{n}(x)}{\sin{(x^{m})}}=0

Problem 9: Find \lim_{x \rightarrow 0}\frac{e^{ax}-e^{bx}}{x}

Problem 10: Find the value of a given the following:

f(x) = - 4x when x less than or equal to -2.

f(x)=a.x.x when x greater than -2.

given that \lim_{x \rightarrow -2} {f(x)} exists.

Problem 11: Let f(x) = \frac{\sin{(e^{x-2}-1)}}{\log{(x-1)}}, then find the value of \lim_{x \rightarrow 2} f(x)

Problem 12: Let f(x) = \frac{1}{\sqrt{(18-x*x)}} then find the value of \lim_{x \rightarrow 3}\frac{f(x)-f(3)}{x-3}

Problem 13: The function f(x) = \frac{\log{(1+ax)-\log{(1-ax)}}}{x} is not defined when x is zero. In order to make this function continuous at zero, what should be the value of f at zero?

Problem 14: If the function given below is continuous at x=3, find the value of c:

f(x) = 3x-5 for x<3

f(x)=x+1 for x>3

f(x)=c for x=3

Problem 15: If f(x) = x+2, when x \leq 1 and f(x)=4x-1, when f(x) = 4x-1 when x>1, then which of the following is true ? (a) f(x) is continuous at x=1 (b) \lim_{x \rightarrow 1}f(x) =4 (c) f(x) is discontinuous at x=4 (d) none of these.

Problem 16:

If \phi(x) = \frac{1-\cos{(\lambda x)}}{x \sin{x}}, when x is not zero, and \phi{(0)} = \frac{1}{2}. If \phi{(x)} is continuous at x=0, then find the value of \lambda.

Problem 17:

Let f(x) = \frac{1-\sin{(x)}}{(\pi-2x)^{2}} where x \neq \frac{\pi}{2} and f(\frac{\pi}{2})=\lambda, then which value of \lambda will make f(x) continuous at x = \frac{\pi}{2}

Regards,

Nalin Pithwa

By Nalin Pithwa | Posted in IITJEE Mains | Comments (0)

Limits and Continuity: IITJEE Maths : Tutorial problems 1

September 26, 2020 – 12:57 pm

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

By Nalin Pithwa | Posted in calculus, IITJEE Mains | Comments (0)