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

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 → -2 ⇒ h → 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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