## Monthly Archives: October 2020

### Derivatives: part 3: IITJEE maths tutorial problems for practice

Problem 1:

Differential coefficient of $\log[10]{x}$ w.r.t. $\log[x]{10}$ is

(a) $\frac{(\log{x})^{2}}{(\log{10})^{2}}$ (b) $\frac{(\log[x]{10})^{2}}{(\log{10})^{2}}$ (c) $\frac{(\log[10]{x})^{2}}{(\log{10})^{2}}$ (d) $\frac{(\log{10})^{2}}{(\log{x})^{2}}$

Problem 2:

The derivative of an even function is always:

(a) an odd function (b) does not exist (c) an even function (d) can be either even or odd.

Problem 3:

The derivative of $\arcsin{x}$ w.r.t. $\arccos{\sqrt{1-x^{2}}}$ is

(a) $\frac{1}{\sqrt{1-x^{2}}}$ (b) $\arccos{x}$ (c) $1$ (d) $\arctan{(\frac{1}{\sqrt{1-x^{2}}})}$

Problem 4:

If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, then $\frac{dy}{dx}$ is

(a) $\frac{\sqrt{1-y^{2}}}{\sqrt{1-x^{2}}}$ (b) $\sqrt{1-x^{2}}$ (c) $\frac{\sqrt{1-x^{2}}}{\sqrt{1-y^{2}}}$ (d) $\sqrt{1-y^{2}}$

Problem 5:

$\frac{d}{dx} \arcsin{2x\sqrt{1-x^{2}}}$ is equal to

(a) $\frac{2}{\sqrt{1-x^{2}}}$ (b) $\cos{2x}$ (c) $\frac{1}{2\sqrt{1-x^{2}}}$ (d) $\frac{1}{\sqrt{1-x^{2}}}$

Problem 6:

If $y=\arctan{\frac{x}{2}}-\arccos{\frac{x}{2}}$, then $\frac{dy}{dx}$ is

(a) $\frac{2}{1+x^{2}}$ (b) $\frac{2}{4+x^{2}}$ (c) $\frac{4}{4+x^{2}}$ (d) $0$

Problem 7:

If $y=\arccos{(\frac{\sqrt{1+\sin{x}}+\sqrt{1-\sin{x}}}{\sqrt{1+\sin{x}}-\sqrt{1-\sin{x}}})}$, then $\frac{dy}{dx}$ is equal to:

(a) $\frac{1}{2}$ (b) $\frac{2}{3}$ (c) $3$ (d) $\frac{3}{2}$

Problem 8:

If $y = \arctan{\frac{4x}{1+5x^{2}}} + \arctan{\frac{2+3x}{3-2x}}$, then $\frac{dy}{dx}$ is

(a) $\frac{1}{1+x^{2}}$ (b) $\frac{5}{1+25x^{2}}$ (c) $1$ (d) $\frac{3}{1+9x^{2}}$

Problem 9:

If $2^{x}+2^{y}=2^{x+y}$, then $\frac{dy}{dx}$ is equal to

(a) $\frac{2^{x}+2^{y}}{2^{x}-2^{y}}$ (b) $2^{x-y} \times \frac{2^{y}-1}{1-2^{x}}$ (c) $\frac{2^{x}+2^{y}}{1+2^{x+y}}$ (d) $\frac{2^{x+y}-2^{x}}{2^{y}}$

Problem 10:

If $y^{2}=p(x)$, a polynomial of degree 3, then $2\frac{d}{dx}(y^{3}\frac{d^{2}y}{dx^{2}})$ is equal to

(a) $p^{'''}(x)+p^{'}(x)$ (b) $p^{''}(x).p^{'''}(x)$ (c) $p^{'''}(x).p(x)$ (d) a constant.

Regards,

Nalin Pithwa.

### Derivatives : part 2: IITJEE Maths : Tutorial problems for practice

Problem 1:

If $f(a)=2$, $f^{'}(a)=1$, $g(a)=-1$, $g^{'}(a)=2$, then the value of $\lim_{x \rightarrow a}\frac{g(x)f(a)-g(a)f(x)}{x-a}$ is

(a) -5 (b) $\frac{1}{5}$ (c) 5 (d) 0

Problem 2:

Let $y = \arcsin{(\frac{2x}{1+x^{2}})}$, $0 < x <1$ and $0 < y < \frac{\pi}{2}$, then $\frac{dy}{dx}$ is equal to :

(a) $\frac{2}{1+x^{2}}$ (b) $\frac{2x}{1+x^{2}}$ (c) $\frac{-2}{1+x^{2}}$ (d) none

Problem 3:

Let $f(x) = ax^{2}+1$ for $x \leq 1$

and $f(x)= x+a$ for $x \leq 1$ then f is derivable at $x=1$, if

(a) $a=0$ (b) $a = \frac{1}{2}$ (c) $a=1$ (d) $a=2$

Problem 4:

If $f(x) = ax^{2}+b$ for $x \leq 1$

if $f(x)=b x^{2}+ax+c$ for $x>1$, where $b \neq 0$, then $f(x)$ is continuous and differentiable at $x=1$, if

(a) $c=0, a=2b$ (b) $a=2b, c \in \Re$ (c) $a=b, c=0$ (d) $a=2b, c \neq 0$

Problem 5:

$\lim_{h \rightarrow 0} \frac{\cos^{2}(x+h)- \cos^{2}(x)}{h}$ is equal to

(a) $\cos^{2}(x)$ (b) $-\sin{2x}$ (c) $\sin{x} \cos{x}$ (d) $2\sin{x}$

Problem 6:

$\lim_{h \rightarrow 0} \frac{\sin{\sqrt{x+h}-\sin{\sqrt{x}}}}{h}$ is equal to

(a) $\cos {\sqrt{x}}$ (b) $\frac{1}{2\sin{\sqrt{x}}}$ (c) $\frac{\cos{\sqrt{x}}}{2\sqrt{x}}$ (d) $\sin{\sqrt{x}}$

Problem 7:

$(\arccos{x})^{'}= \frac{-1}{\sqrt{1-x^{2}}}$ where

(a) $-1 < x <1$ (b) $-1 \leq x \leq 1$ (c) $-1 \leq x < 1$ (d) $-1 < x \leq 1$

Problem 8:

$\frac{d}{dx}(\arctan{(\frac{3x-x^{2}}{1-3x^{2}})})$ is equal to

(a) $\frac{3}{1+x^{2}}$ (b) $\frac{3}{1+9x^{2}}$ (c) $\sec^{2}{x}$ (d) $\frac{1}{9+x^{2}}$

Problem 9:

If $x=a\cos^{3}(t)$ and $y=a\sin^{3}(t)$, then $\frac{dy}{dx}$ is equal to

(a) $\cos{t}$ (b) $\cot{t}$ (c) $cosec{(t)}$ (d) $-\tan{t}$

Problem 10:

If $y = arcsin{\cos{x}}$, then $\frac{dy}{dx}$ is equal to

(a) -1 (b) $\cos{t}$ (c) $cosec{(t)}$ (d) $-\tan{t}$

Regards,

Nalin Pithwa

### Derivatives: part 1: IITJEE Maths Tutorial Problems Practice

Problem 1:

If $y=x^{x}$, $x>0$, then find $\frac{dy}{dx}$.

Problem 2:

If $y= x^{x^{x^{\ldots}}}$, then find the value of $x\frac{dy}{dx}$.

Problem 3:

Find the derivative of $e^{\ln{x}}$ w.r.t. x.

Problem 4:

Let $f(x) = \log{(x+\sqrt{x^{2}+1})}$, then find the value of $f^{'}(x)$.

Problem 5:

If $y= \arctan{\frac{\sqrt{1+x^{2}}-1}{x}}$, then find the value of $y^{'}(0)$.

Problem 6:

If $y=t^{2}+t-1$, then find the value of $\frac{dy}{dx}$.

Problem 7:

If $x=a(t-\sin{t})$, $y=a(1+\cos{t})$, then evaluate $\frac{dy}{dx}$.

Problem 8:

If $x^{y}=e^{x-y}$, then evaluate $\frac{dy}{dx}$.

Problem 9:

If $y= \sec^{-1}{(\frac{x+1}{x-1})} + \arcsin{(\frac{x-1}{x+1})}$, then evaluate $\frac{dy}{dx}$.

Problem 10:

If $y = \arctan{(\frac{\sin{x}+\cos{x}}{\cos{x}-\sin{x}})}$, then find $\frac{dy}{dx}$

Problem 11:

If $\sqrt{x}+\sqrt{y}=4$, then evaluate $\frac{dy}{dx}$ at $y=1$.

Problem 12:

If $f(x) = \frac{x-4}{2\sqrt{x}}$, then evaluate $f^{'}(0)$.

Problem 13:

If $f^{'}(x) = \sin{\log{x}}$ and $y=f(\frac{2x+3}{3-2x})$, find $\frac{dy}{dx}$. One of the given choices is correct:

(a) $\frac{12\cos{(\log{x})}}{x(3-2x)^{2}}$

(b) $\frac{12\sin{\log{(\frac{2x+3}{3-2x})}}}{(3-2x)^{2}}$

(c) $\frac{12\cos{\log{(\frac{2x+3}{3-2x})}}}{x(3-2x)^{2}}$

(d) none of these

Problem 14:

If $f(0)=0=g(0)$ and $f^{'}(0)=6=g^{'}(0)$, then $\lim_{x \rightarrow 0} \frac{f(x)}{g(x)}$ is given by:

(a) 1 (b) 0 (c) 12 (d) -1

Regards,

Nalin Pithwa

### 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