Question

If $f\left(x\right)=\{\begin{array}{cc}\frac{8^x-4^x-2^x+1}{x^2},& x>0\\ e^x\sin x+\pi x+\lambda \ln 4,& x\le 0\end{array}$ is continuous at $x=0$, then the value of $\lambda$ is

• A. $4{\log }_{e}2$
• B. $2{\log }_{e}2$
• C. ${\log }_{e}2$
• D. none of these

Answer 1

The correct option is C ${\log }_{e}2$

Given f(x) is continuous at $x=0$
$\Rightarrow LHL=RHL=f\left(0\right)$
Now, $f\left(0\right)=0+0+\lambda ln4=\lambda ln4$ ....(1)
$R.H.L.=\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(0+h)$

$\underset{h\to 0}{lim}\frac{8^h-4^h-2^h+1}{h^2}$

$\underset{h\to 0}{lim}\frac{(4^h-1)(2^h-1)}{h.h}$

$\underset{h\to 0}{lim}\left(\frac{4^h-1}{h}\right)\underset{h\to 0}{lim}\left(\frac{2^h-1}{h}\right)$
$\Rightarrow RHL=\log 4\log 2$ ..... (2)
$\therefore f\left(0\right)=R.H.L.$
$\therefore \lambda =\log 2$