## Limits, Continuity and Differentiability Questions and Answers Part-7

1. The value of $\lim_{x \rightarrow 0}\frac{\sqrt{x^{2}+1}-1}{\sqrt{x^{2}+9}-3}$      is
a) 4
b) 3
c) 1
d) 2

Explanation:

2. The value of $\lim_{x \rightarrow b}\frac{\sqrt{x-a-}\sqrt{b -a}}{x^{2}-b^{2}}\left(b> a\right)$      is
a) $\frac{1}{4b}$
b) $\frac{1}{b\sqrt{b-a}}$
c) $\frac{1}{2b\sqrt{b-a}}$
d) $\frac{1}{4b\sqrt{b-a}}$

Explanation:

3. If $f\left(x\right)=\frac{\log\left(e^{x^{2}}+2\sqrt{x}\right)}{\tan\sqrt{x}},x\neq0.$
The value of $\lim_{x \rightarrow 0}f\left(0\right)$   is
a) $\frac{1}{2}$
b) $\sqrt{2}$
c) 2
d) $\frac{1}{\sqrt{2}}$

Explanation:

4. The value of $\lim_{x \rightarrow \infty}\frac{\left(x+1\right)^{20}+\left(x+2\right)^{20}+....+\left(x+100\right)^{20}}{x^{20}+10^{20}}$
is
a) 100
b) 1
c) 10
d) 20

Explanation:

5. If $f\left(x\right)=\left(\frac{x^{2}+4x+3}{x^{2}+x+2}\right)^{x}$     , then $\lim_{x \rightarrow \infty}f\left(x\right)$   is
a) $e^{3}$
b) $e^{4}$
c) $e^{2}$
d) $2^{4}$

Explanation:

6. Let $f\left(x\right)=\left(\tan \left(\pi/4-x\right)/\cot 2x\right)\left(x\neq\pi/4\right).$
The value which should be assigned to f at $x=\pi/4$ , so that it is continuous everywhere , is
a) 1/2
b) 1
c) 2
d) 1/4

Explanation:

7. Let $f\left(x\right)=\frac{x \tan 2x-2x\tan x}{\left(1-\cos 2x\right)^{2}},x\neq0$
Then the value f(0) so that f is continuous
a) 2
b) -2
c) 1/2
d) -1/2

Explanation:

8. $\lim_{x \rightarrow 1}\frac{1+\log x-x}{1-2x+x^{2}}$     equals
a) 1
b) 0
c) -1
d) -1/2

Explanation:

9. Let f be a continuous function satisfying f(x) f(y) = f (x) + f(y) + f (xy) – 2 for all x, $y\epsilon R$  and f(2) = 5 then $\lim_{x \rightarrow 4}f\left(x\right)$   is
a) 5
b) 17
c) -5
d) 21

10. The value of $\lim_{x \rightarrow 0}\left(\frac{2+\cos x}{x^{3}\sin x}-\frac{3}{x^{4}}\right)$     is
a) $\frac{1}{30}$
c) $\frac{1}{60}$