## 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

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