## Limits, Continuity and Differentiability Questions and Answers Part-5

1. Let $f:R\rightarrow R$   be given by f(x) = 5x , if $x\epsilon Q$  and $f\left(x\right)=x^{2}+6$    if $x\epsilon R \sim Q$   then
a) f is continuous at x=1 and x=2
b) f is not continuous at x=1 and x=2
c) f is continuous at x=1 but not at x=2
d) f is continuous at x=2 but not at x=1

Explanation:

2. If $\lim_{x \rightarrow 0}\frac{\left(\left(a-n\right)nx-\tan x\right)\sin n x}{x^{2}}=0$
where $n\epsilon R \sim\left\{0\right\}$    then a is equal to
a) 0
b) $\frac{n}{n+1}$
c) n
d) n+1/n

Explanation:

3. Let $f\left(x\right)=\lim_{n \rightarrow \infty} \frac{x}{x^{2n}+1}$    then
a) $\lim_{x \rightarrow 1+} f\left(x\right)+\lim_{x \rightarrow 1-} f\left(x\right)=1$
b) $\lim_{x \rightarrow 1+} f\left(x\right)+\lim_{x \rightarrow 1-} f\left(x\right)+f\left(1\right)=\frac{3}{2}$
c) $\lim_{x \rightarrow 1+} f\left(x\right)+\lim_{x \rightarrow 1-} f\left(x\right)=0$
d) All of the Above

Explanation:

4. If $f\left(x\right)=\left(\frac{x}{3+x}\right)^{x}$  , then
a) $\lim_{x \rightarrow \infty} f\left(x\right)=e^{-3}$
b) $\lim_{x \rightarrow 1} f\left(x\right)=1/4$
c) $\lim_{x \rightarrow \infty} f\left(x\right)=e^{-5}$
d) Both a and b

Explanation:

5. If $f\left(x\right)=x\frac{e^{\left[x\right]+\mid x\mid}-2}{\left[x\right]+\mid x\mid}$    then
a) $\lim_{x \rightarrow 0+}f\left(x\right)=-1$
b) $\lim_{x \rightarrow 0-}f\left(x\right)=0$
c) $\lim_{x \rightarrow 0}f\left(x\right)$   does not exist
d) All of the Above

Explanation:

6. The $\lim_{x \rightarrow 0}x^{8}\left[\frac{1}{x^{3}}\right]$    ( where [x] is greatest integer function) is
a) zero
b) a rational number
c) an integer
d) All of the Above

Explanation:

7. If $f\left(x\right)=\lim_{n \rightarrow\infty}\left(\sin x\right)^{2n}$    , then f is
a) continuous at $x =\pi$
b) discontinuous at $x =\pi/2$
c) discontinuous at $x =-\pi/2$
d) All of the Above

Explanation:

8. Let f(x) be defined on $\left[0,\pi\right]$   by
$f\left(x\right)=\begin{cases}x+a\sqrt{2}\sin x & ,0\leq x\leq\pi/4\\2x\cot x+b&\ ,\pi/4< x\leq\pi/2 \\a\cos2x-b\sin x &\ ,\pi/2< x\leq\pi \end{cases}$
If f is continuous on $\left[0,\pi\right]$   then
a) $a=\pi/6$
b) $b=-\pi/12$
c) $a=\pi/6$   and $b-\pi/12$
d) All of the Above

Explanation:

9. Let $f\left(x\right)=g'\left(x\right)\frac{e^{a/x}-e^{-a/x}}{e^{a/x}+e^{-a/x}}$     where g' is the derivative of g and is a continuous function and a> 0 then $\lim_{x \rightarrow 0}f(x)$   exist if
a) g(x) is polynomial
b) $g\left(x\right)=x^{3}$  h(x) where h(x) is a polynomial
c) $g\left(x\right)=x^{2}$
d) Both b and c

10. Let $f\left(x\right)=\left[x\right]+\left[-x\right]$     . then for any integer n and $k \epsilon R \sim I$
a) $\lim_{x \rightarrow n}f\left(x\right)$   exists
b) $\lim_{x \rightarrow k}f\left(x\right)$   exists