If f(x) = begin{cases} x left( frac{e^{1/x} - | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) First,check continuity at $x = 0$: $\lim_{x 	o 0} f(x) = \lim_{x 	o 0} x \left( \frac{e^{1/x} - e^{-1/x}}{e^{1/x} + e^{-1/x}} ight)$. Since $\

Vedclass · Accepted Answer

$f$ is differentiable at every point

Vedclass · Answer

(C) First,check continuity at $x = 0$: $\lim_{x 	o 0} f(x) = \lim_{x 	o 0} x \left( \frac{e^{1/x} - e^{-1/x}}{e^{1/x} + e^{-1/x}} ight)$. Since $\left| \frac{e^{1/x} - e^{-1/x}}{e^{1/x} + e^{-1/x}} ight| Now,check differentiability at $x = 0$: $LHD = \lim_{h 	o 0^+} \frac{f(0-h) - f(0)}{-h} = \lim_{h 	o 0^+} \frac{-h \left( \frac{e^{-1/h} - e^{1/h}}{e^{-1/h} + e^{1/h}} ight)}{-h} = \lim_{h 	o 0^+} \frac{e^{1/h} - e^{-1/h}}{e^{1/h} + e^{-1/h}} = \lim_{h 	o 0^+} \frac{1 - e^{-2/h}}{1 + e^{-2/h}} = \frac{1-0}{1+0} = 1$. $RHD = \lim_{h 	o 0^+} \frac{f(0+h) - f(0)}{h} = \lim_{h 	o 0^+} \frac{h \left( \frac{e^{1/h} - e^{-1/h}}{e^{1/h} + e^{-1/h}} ight)}{h} = \lim_{h 	o 0^+} \frac{1 - e^{-2/h}}{1 + e^{-2/h}} = 1$. Since $LHD = RHD = 1$,the function is differentiable at $x = 0$. Thus,$f$ is differentiable at every point.

If $f(x) = \begin{cases} x \left( \frac{e^{1/x} - e^{-1/x}}{e^{1/x} + e^{-1/x}} \right), & x \neq 0 \\ 0, & x = 0 \end{cases}$,then the correct statement is:

Similar Questions

The function $f(x)=|x^{2}-2 x-3| \cdot e^{|9 x^{2}-12 x+4|}$ is not differentiable at exactly :

If the function $f(x) = \begin{cases} -x, & x < 1 \\ a + \cos^{-1}(x + b), & 1 \le x \le 2 \end{cases}$ is differentiable at $x = 1$,then $\frac{a}{b}$ is equal to:

If $f(x) = \begin{cases} \tan^{-1} x, & \text{when } |x| \leq 1 \\ \frac{1}{2}(|x|-1), & \text{when } |x| > 1 \end{cases}$,then the domain of $\frac{d}{dx} f(x)$ is

Which of the following functions is differentiable at $x = 0$?

The function $f(x) = x^2 \sin \frac{1}{x}$ for $x \ne 0$ and $f(0) = 0$ at $x = 0$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module