## Limits, Continuity and Differentiability Questions and Answers Part-9

1. The set of all points where the function f (x) = | x | sin x is differentiable is
a) $\left(-\infty,\infty\right)$
b) $\left(-\infty,0\right)\cup\left(0,\infty\right)$
c) $\left(0,\infty\right)$
d) $\left[0,\infty\right)$

Explanation:

2. If the function f is differentiable and strictly increasing in a neighborhood of 0, then $\lim_{x \rightarrow 0}\left[\frac{f\left(x^{4}\right)-f\left(x^{2}\right)}{f\left(x^{2}\right)-f\left(0\right)}+\frac{f\left(x^{3}\right)-f\left(x\right)}{f\left(x\right)-f\left(0\right)}\right]$
is equal to
a) -1
b) -2
c) 0
d) 1

Explanation: As f is strictly increasing so f'(x) > 0 for all x in a neighborhood of 0

3. The set of all points where the function $f\left(x\right)=\sqrt{1-e^{-x^{2}}}$     is differentiable is
a) $\left(0,\infty\right)$
b) $\left(-\infty,\infty\right)$
c) $\left(-\infty,\infty\right)\sim\left\{0\right\}$
d) $\left(-1,\infty\right)$

Explanation: For x $\neq$ 0, we have

4. For $n\epsilon N$ , let $f\left(x\right)=\min\left\{1-\tan^{n}x,1-\sin^{n} x, 1-x^{n}\right\} ,x\epsilon \left(-\pi/2,\pi/2\right).$
The left hand derivative of f at $x =\pi/4$  is
a) -2n
b) -2(n+1)
c) $-n\frac{\pi}{4}$
d) $-n\left(\frac{\pi}{4}\right)^{n-1}$

Explanation:

5. If f (a) = 2, f '(a) = 1, g(a) = –1 and g'(a) = 2, the value of $\lim_{x \rightarrow a}\frac{g\left(x\right)f\left(a\right)-g\left(a\right)f\left(x\right)}{x-a}$
is
a) -5
b) 1/5
c) 5
d) 2/5

Explanation:

6.Let $g_{n}\left(x\right)=\begin{cases}x^{n}\sin\left(\frac{1}{x}\right) & x\neq 0\\0 & x = 0\end{cases}$
then
a) $g_{1}$  is differentiable at x = 0
b) $g_{0}$  is continuous at x = 0
c) $g_{2}$  is continuously differentiable
d) $g_{3}$  is continuously differentiable

Explanation:

7. If f' is differentiable function and f" (x) is continuous at x = 0 and f" (0) = 5, the value of $\lim_{x \rightarrow 0}\frac{2f\left(x\right)-3f\left(2x\right)+f\left(4x\right)}{x^{2}}$
is
a) 5
b) 10
c) 15
d) 20

Explanation:

8. Let [] denote the greatest integer function and $f\left(x\right)=\left[\tan^{2}x\right]$   . then
a) $\lim_{x \rightarrow 0}f\left(x\right)$   does not exist
b) f (x) is continuous at x = 0
c) f (x) is not differentiable at x = 0
d) f'(0) = 1

Explanation:

9. Let f (x + y) = f (x) f ( y) for all x and y. If f (5) = 2 and f'(0) = 3, then f'(5) is equal to
a) 5
b) 6
c) 0
d) 3

10. Let f(x) = [x] and $g\left(x\right)=\begin{cases}0 &x= integer\\x^{2} & otherwise \end{cases}$