## Limits, Continuity and Differentiability Questions and Answers Part-4

1. If $f\left(x\right)=\begin{cases}\left(1+\mid\sin x\mid\right)^{a/\mid\sin x\mid} & - \pi/6< x < 0\\b&\ x=0 \\e^{\tan 2x/\tan 3x} &\ 0< x< \pi/6 \end{cases}$
is a continuous function on $\left(-\pi/6,\pi/6\right)$   , then
a) $a = 2/3, b = e^{2}$
b) $a = 1/3, b = e^{1/3}$
c) $a = 2/3, b = e^{2/3}$
d) None of these

Explanation:

2. Let $f\left(x\right)=\lim_{n \rightarrow \infty}\frac{\log\left(2+x\right)-x^{2n}\sin x}{1+x^{2n}},$
then
a) f is continuous at x=1
b) $\lim_{x \rightarrow1+}f\left(x\right)\neq \lim_{x \rightarrow1-}f\left(x\right)$
c) $\lim_{x \rightarrow1+}f\left(x\right)=\sin 1$
d) $\lim_{x \rightarrow1-}f\left(x\right)$    doesn't exist

Explanation:

3. Let $f\left(x\right)=g\left(x\right)\frac{e^{1/x}-e^{-1/x}}{e^{1/x}+^{-1/x}}$     and $x\neq0$  where g is a continuous function. Then $\lim_{x \rightarrow 0}$  f(x) exists if
a) g(x) is any polynomial
b) g(x) = x + 4
c) $f\left(x\right)=x^{2}$
d) $f\left(x\right)=2+3x+4x^{2}$

Explanation:

4. The value of $\lim_{n \rightarrow \infty } \cos\frac{x}{2}\cos\frac{x}{4}....\cos\frac{x}{2^{n}}$      is
a) 1
b) $\frac{\sin x}{x}$
c) $\frac{x}{\sin x}$
d) None of these

Explanation:

5. Let f be continuous function on R such that $f\left(\frac{1}{4n}\right)=\left(\sin e^{n}\right)e^{-n^2}+\frac{n^{2}}{n^{2}+1}.$
Then the value of f(0) is
a) 1
b) 1/2
c) 0
d) None of these

Explanation:

6. Let $f\left(x\right)=\left\{\frac{\log \left(1+x\right)^{1+x}}{x^{2}}-\frac{1}{x}\right\}$
Then the value of f(0) so that the function f is continuous is
a) 1
b) 1/2
c) 2/4
d) None of these

Explanation:

7. If $\lim_{x \rightarrow a}(f\left(x\right)g\left(x\right))$   exists for any functions f and g then
a) $\lim_{x \rightarrow a}f\left(x\right)$   and $\lim_{x \rightarrow a}g\left(x\right)$   exist
b) $\lim_{x \rightarrow a}f\left(x\right)$   exist but $\lim_{x \rightarrow a}g\left(x\right)$   may not exist
c) $\lim_{x \rightarrow a}f\left(x\right)$   mayn't exist but $\lim_{x \rightarrow a}g\left(x\right)$   exist
d) $\lim_{x \rightarrow a}f\left(x\right)$   and $\lim_{x \rightarrow a}g\left(x\right)$   may not exist

Explanation:

8. The value of $\lim_{x \rightarrow 0}\frac{\left(1+x\right)^{1/x}-e+\frac{1}{2}ex}{x^{2}}$     is
a) $\frac{11}{24}e$
b) $-\frac{11}{24}e$
c) $\frac{e}{24}$
d) None of these

Explanation:

9. The value of $\lim_{x \rightarrow 0}\frac{x^{2}\sin1/x}{\sin x}$     is
a) 1
b) 0
c) 1/2
d) None of these

10. The value of $\alpha$  so that $\lim_{x \rightarrow 0}\frac{1}{x^{2}}\left(e^{\alpha x}-e^{x}-x\right)=\frac{3}{2}$       is