## 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

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