Name: Differentiation of Exponential Functions
Uploaded: 2019-07-19
Duration: 12 min 3 s
Description: To know how to find the derivative of exponential functions or multiple functions containing the exponential term. Let's watch this video and explore!

Question

Show that the function

f (x) = 2 x - |x|

is continuous at

x = 0

but not differentiable at

x = 0

.

Accepted Answer

Given function is $f (x) = 2 x - |x|$ ,

We know that $|x| = \{\begin{cases} x, & i f x \geq 0 \\ - x, & i f x < 0 \end{cases}$ [The absolute value or modulus of a real number $x$ , denoted by $|x|$ , is the non-negative value of $x$ without regard to its sign. Namely $|x| = x$ , if $x$ is positive; $|x| = - x$ , if $x$ is negative and $|0| = 0$ ]

We have to show that the function is continuous at $x = 0$ but not differentiable at $x = 0$ .

Continuity at $x = 0$ :

Here the domain of the function is $D = R$ .

$LHL = \lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{-}} (2 x - |x|) = \lim_{x \to 0^{-}} (2 x - (- x)) [∵ a s x \to 0^{-} \Rightarrow x < 0; ∴ |x| = - x]$

$= \lim_{x \to 0^{-}} 3 x = 3 \times 0 = 0$

$RHL = \lim_{x \to 0^{+}} f (x) = \lim_{x \to 0^{+}} (2 x - |x|) = \lim_{x \to 0^{+}} (2 x - x) [∵ a s x \to 0^{+} \Rightarrow x > 0; ∴ |x| = x]$

$= \lim_{x \to 0^{+}} x = 0$

Also $f (0) = 2 \times 0 - |0| = 0$

$∴ LHL = RHL = f (0)$

Thus, $f (x)$ is continuous at $x = 0$ .

Differentiability at $x = 0$ :

$L f' (0) = \lim_{h \to 0} \frac{f (0 - h) - f (0)}{- h} = \lim_{h \to 0} \frac{(2 (0 - h) - |0 - h|) - 0}{- h} = \lim_{h \to 0} \frac{(- 2 h - (h)) - 0}{- h} = \lim_{h \to 0} \frac{(- 2 h - h) - 0}{- h}$

$= \lim_{h \to 0} \frac{- (2 h + h)}{- h} = \lim_{h \to 0} \frac{3 h}{h} = \lim_{h \to 0} 3 = 3$

$R f' (0) = \lim_{h \to 0} \frac{f (0 + h) - f (0)}{h} = \lim_{h \to 0} \frac{(2 h - |h|) - 0}{h} = \lim_{h \to 0} \frac{(2 h - h) - 0}{h}$

$= \lim_{h \to 0} \frac{h}{h} = 1$

$∴ L f' (0) \neq R f' (0)$

Thus, $f (x)$ is not differentiable at $x = 0$ .

Hence, $f (x)$ is continuous at $x = 0$ but not differentiable at $x = 0$ .

Show that the function $f (x) = 2 x - |x|$ is continuous at $x = 0$ but not differentiable at $x = 0$ .

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Show that the function fx=2x-x is continuous at x=0 but not differentiable at x=0.

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Show that the function $f (x) = 2 x - |x|$ is continuous at $x = 0$ but not differentiable at $x = 0$ .