If f (x)= ll 3 x e ^ 2 x -1 & ; x 0 k -2 & ; x=0 . is continuous … - Vidyadip

MCQ

If $f (x)=\left\{\begin{array}{ll}\frac{\sin 3 x}{ e ^{2 x}-1} & ; x \neq 0 \\ k -2 & ; x=0\end{array}\right.$ is continuous at $x=0$, then $k =$

A
$\frac{9}{5}$
B
$\frac{2}{3}$
✓
$\frac{7}{2}$
D
$\frac{1}{2}$

✓

Answer

Correct option: C.

$\frac{7}{2}$

(C)
f is continuous at $x=0$
$\Rightarrow \lim _{x \rightarrow 0} f(x)=f(0)$
$\Rightarrow \lim _{x \rightarrow 0} \frac{\sin 3 x}{ e ^{2 x}-1}= k -2$
$\Rightarrow \lim _{x \rightarrow 0} \frac{\frac{\sin 3 x}{x}}{\frac{ e ^{2 x}-1}{x}}= k -2$
$\Rightarrow \frac{3 \lim _{x \rightarrow 0} \frac{\sin 3 x}{3 x}}{2 \lim _{x \rightarrow 0} \frac{ e ^{2 x}-1}{2 x}}= k -2$
$\Rightarrow \frac{3}{2}= k -2$
$\Rightarrow k=\frac{7}{2}$

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