Question

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

• A. $\frac{7}{2}$
• B. $\frac{3}{2}$
• C. $\frac{2}{3}$
• D. $\frac{9}{5}$

Answer 1

Answer: (A)

$f(x)=\{\begin{array}{ll}\frac{sin3x}{e^{2x}-1};& x\ne 0\\ k-2;& x=0\end{array}$
Since f is continuous at x = 0
$\Rightarrow li{m}_{x\to 0}f\left(x\right)=f\left(0\right)$
$li{m}_{x\to 0}\frac{sin3x}{e^{2x}-1}=k-2$
$\Rightarrow \frac{li{m}_{x\to 0}\frac{sin3x\times 3x}{3x}}{li{m}_{x\to 0}\frac{e^{2x}-1\times 2x}{2x}}=k-2$
$\Rightarrow \frac{3}{2}=k-2$
$\Rightarrow k=\frac{3}{2}+2=\frac{7}{2}$
$\therefore k=\frac{7}{2}$