Mathematics Hothouse

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

October 18, 2020 – 8:00 am

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

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

Wisdom of George Polya

October 17, 2020 – 6:50 am

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

By Nalin Pithwa | Posted in mathematicians, memory power concentration retention, miscellaneous, motivational stuff | Comments (0)

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

October 16, 2020 – 3:24 pm

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 $\frac{\pi}{4}$

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

(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

(d) none

Regards,

Nalin Pithwa

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

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

October 16, 2020 – 1:28 pm

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) $x^{2} + 3x +2=0$

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 $x=0$

(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

(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

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

Regards,

Nalin Pithwa

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

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