## Limits, Continuity and Differentiability Questions and Answers Part-1

1. $\lim_{x \rightarrow 0}\left(\frac{1-\cos^{5}x}{1-\cos^{2}x}\right)$     is equal to
a) $\frac{3}{2}$
b) $\frac{5}{2}$
c) 1
d) 2

Explanation:

2. Let f(x)=$\begin{cases}\frac{\frac{3}{x+1}+\sqrt{x+11}}{x+2}, & x \neq -2\\k, & x = -2\end{cases}$
The value of k so that f is a continuous at x =-2 is
a) 2
b) $\frac{11}{4}$
c) $\frac{17}{4}$
d) $-\frac{17}{6}$

Explanation:

3. For a, b > 0 , the value of $\lim_{n \rightarrow \infty}\left(4\left(\frac{a^{\frac{1}{n+1}}+b^{\frac{1}{n+1}}}{2}\right)^{n+1}-\left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^{n}\right)$
is equal to
a) $3\sqrt{ab}$
b) $\frac{1}{2}\sqrt{ab}$
c) $\sqrt{a}+\sqrt{b}$
d) $\sqrt{a}-\sqrt{b}$

Explanation:

4. The value of $\lim_{x \rightarrow 0}\log_{\tan^{2}x}\left(\tan^{2}2x\right)$      is
a) 2
b) 0
c) 1
d) $\frac{1}{2}$

Explanation:

5. Let $f\left(x\right)= \begin{cases}x+a & if & x < 0\\\mid x-1\mid & if & x \geq 0\end{cases}$
and $g\left(x\right)= \begin{cases}x+1 & if & x < 0\\\left( x-1\right)^{2}+b& if & x \geq 0\end{cases}$
The value of (a,b) so that gof is continuous
a) (1,1)
b) (2,1)
c) (1,2)
d) (1,0)

Explanation:

6. $\lim_{x \rightarrow 0}\frac{\tan x-\sin x}{x^{3}}$     is
a) 0
b) 1/2
c) 2
d) none of these

Explanation:

7. $\lim_{n \rightarrow \infty}\left(\frac{1}{1-n^{2}}+\frac{2}{1-n^{2}}+....+\frac{n}{1-n^{2}}\right)$        is equal to
a) 0
b) -1/2
c) 1/2
d) none of these

Explanation:

8. If $f\left(x\right)=\begin{cases}\frac{\left[x\right]^{2}+\sin\left[x\right]}{\left[x\right]} & for\left[x\right] \neq 0\\0 & for \left[x\right] = 0\end{cases}$
where [x] denoted the greatest integer less than or equal to x, then $\lim_{x \rightarrow 0}f\left(x\right)$   equal
a) 1
b) 0
c) -1
d) $\lim_{x \rightarrow 0+}f(x) =0$

Explanation:

9. $\lim_{x \rightarrow \pi/3}\frac{\sin\left(\pi/3-x\right)}{2\cos x-1}$      is equal to
a) 1/2
b) $1/\sqrt{3}$
c) $\sqrt{3}$
d) $2/\sqrt{3}$

10. $\lim_{x \rightarrow0}\frac{1-\cos^{3}x}{x \sin x \cos x}$      is equal to