Discuss the continuity and differentiability of f(x)=e^ |x| . - Vidyadip

Question

Discuss the continuity and differentiability of $\text{f(x)}=\text{e}^{|\text{x}|}.$

✓

Answer

Given:
$\text{f(x)}=\text{e}^{|\text{x}|}$
$\Rightarrow\text{f(x)}=\begin{cases}\text{e}^\text{x},&\text{x}\geq0\\\text{e}^{-\text{x}},&\text{x}<0\end{cases}$
f is Continuity:
(LHL at x = 0)
$\lim_\limits{\text{x}\rightarrow0{^-}}\text{f(x)}$
$=\lim_\limits{\text{h}\rightarrow0}\text{f}(0-\text{h})$
$=\lim_\limits{\text{h}\rightarrow0}\text{e}^{-(0-\text{h})}$
$=\lim_\limits{\text{h}\rightarrow0}\text{e}^{-\text{h}}$
$=1$
(RHL at x = 0)
$\lim_\limits{\text{x}\rightarrow0{^{+}}}\text{f(x)}$
$=\lim_\limits{\text{h}\rightarrow0}\text{f}(0+\text{h})$
$=\lim_\limits{\text{h}\rightarrow0}\text{e}^{(0+\text{h})}$
$=1$
and f(0)
$=\text{e}^0=1$
Thus, $\lim_\limits{\text{x}\rightarrow0^{-}}-\text{f(x)}=\lim_\limits{\text{h}\rightarrow0^{+}}-\text{f(x)}=\text{f(0)}$
Hence, function is continuous at x = 0.
Differentiability at x = 0.
(LHL at x = 0)
$=\lim_\limits{\text{x}\rightarrow0^{-}}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(0-\text{h})-\text{f}(0)}{0-\text{h}-0}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{e}^{-(0-\text{h})}-1}{-\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{e}^\text{h}-1}{-\text{h}}=-1\ \Big[\because\lim_\limits{\text{h}\rightarrow0}\frac{\text{e}^\text{h}-1}{\text{h}}=1\Big]$
(RHL at x = 0)
$=\lim_\limits{\text{x}\rightarrow0^{+}}\frac{\text{f(x)}-\text{f}(0)}{\text{x}-0}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{f}(0-\text{h})-\text{f}(0)}{0+\text{h}-0}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{e}^{-(0-\text{h})}-1}{\text{h}}$
$=\lim_\limits{\text{h}\rightarrow0}\frac{\text{e}^\text{h}-1}{\text{h}}=1\ \Big[\because\lim_\limits{\text{h}\rightarrow0}\frac{\text{e}^\text{h}-1}{\text{h}}=1\Big]$
LHL at (x = 0) $\neq$ RHL at (x = 0)
Hence the function is not differentiable at x = 0.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Differentiability→Differentiability — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Three shopkeepers $\text{A, B}$ and $C$ go to a store to buy stationery. A purchases $12$ dozen notebooks, $5$ dozen pens and $6$ dozen pencils. $B$ purchases $10$ dozen notebooks, $6$ dozen pens and $7$ dozen pencils. $C$ purchases $11$ dozen notebooks, $13$ dozen pens and $8$ dozen pencils. A notebook costs $40$ paise, a pen costs ₹ $1.25$ and a pencil costs $35$ paise. Use matrix multiplication to calculate each individual's bill.

Let, X denote the number of colleges where you will apply after your results and P (X = x) denotes your probability of getting admission in x number of colleges. It is given that
$ \text{P(x = }x) = \begin{cases} \text{K}x\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }, & \text{if } x = 0 \text{ or 1}\\ 2\text{k}x\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }, & \text{if } x = 2\\ \text{k} (5 - x)\text{ }\text{ }\text{ }\text{ }, &\text{if } x = 3 \text{ or 4'}\\ 0\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }, & \text{if } x > 4 \end{cases}$
where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.

Evaluate the following integrals:
$\int\frac{\text{e}^{\text{x}}}{\text{x}}\Big\{\text{x}(\log\text{x})^2+2\log\text{x}\Big\}\text{dx}$

Integrate the function $\frac{x+3}{x^{2}-2 x-5}$

Evaluate the following integrals as limit of sum:
$\int\limits^{4}_{1}\big(\text{x}^2-\text{x}\big)\text{dx}$

$\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}\text{and }\text{B}=\begin{bmatrix}7&2&-6\\ -2&1&-3\\ -4&2&5\end{bmatrix}$, find AB. Hence, solve the system of equations:
x - 2y = 10, 2x + y + 3z = 8 and -2y + z = 7

Evaluate the following integrals: $\int\frac{1}{\sin\text{x}(3+2\cos\text{x})}\ \text{dx}$

Evaluate the following integrals:
$\int_\limits{\frac{1}{3}}^{1}\frac{\big(\text{x}-\text{x}^3\big)^{\frac{1}{3}}}{\text{x}^4}\text{ dx}$

Differentiate the following functions with respect to x:
$\sin(\text{x}^\text{x})$

Find the area of the region bounded by the parabola $y^2 = 2x$ and the straight line $x - y = 4$