If f(x)= |x+2| ^ -1 (x+2) , & x -2 2, & x=-2 , then f(x) is: - Vidyadip

MCQ

If $\text{f(x)}=\begin{cases}\frac{|\text{x}+2|}{\tan^{-1}(\text{x}+2)}, & \text{x}\neq-2\\2, & \text{x}=-2\end{cases},$ then $f(x)$ is:

A
Continuous at $x = -2$
✓
Not continuous at $x = -2$
C
Diffrentiable at $x = -2$
D
Continuous but nit derivable at $x = -2$

✓

Answer

Correct option: B.

Not continuous at $x = -2$

$\lim\limits_{\text{x}\rightarrow-2}\frac{|\text{x}+2|}{\tan^{-1}(\text{x}+2)}$
Let, $x = -2 + h$
$x \rightarrow -2$
$\Rightarrow h \rightarrow 0$
$\lim\limits_{\text{h}\rightarrow0}\frac{|-2+\text{h}+2|}{\tan^{-1}(-2+\text{h}+2)}$
$\lim\limits_{\text{h}\rightarrow0}\frac{\text{h}}{\tan^{-1}\text{h}}=1$
$\lim\limits_{\text{h}\rightarrow0}\frac{|\text{x}+2|}{\tan^{-1}(\text{x}+2)}\neq\text{f}(-2)$
Function is not continuous at $x = -2.$

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