Question

Let $f$ be the function defined as $f\left(x\right)=\{\begin{array}{cc}\frac{{\cos }^{2}x-{\sin }^{2}x-1}{\sqrt{x+1}-1}& ,\text{if}x\ne 0\\ k& ,\text{if}x=0\end{array},$ if $x\ne 0,$ is continuous at $x=0$
Find the value of k

Answer 1

$RHL=\underset{h\to o}{lim}f(x+h)$

$RHL=\underset{h\to o}{lim}f(o+h)$
$=\underset{h\to o}{lim}\left(\frac{co{s}^{2}h-si{n}^{2}h-1}{\sqrt{h+1}-1}\right)$
$=\underset{h\to o}{lim}\left(\frac{1-si{n}^{2}h-si{n}^{2}h-1}{\sqrt{h+1}-1}\right)\cdot \left(\frac{\sqrt{h+1}+1}{\sqrt{h+1}+1}\right)$
$=\underset{h\to o}{lim}\left(\frac{-2si{n}^{2}h(\sqrt{h+1}+1)}{h+1-1}\right)$
$=\underset{h\to o}{lim}-2\left(\left(\frac{sinh}{h}\right)sinh\right(\sqrt{h+1}+1\left)\right)$

$=-2\cdot 1\cdot 0\cdot 2=0$
$\therefore forcontinuityLHL=RHL=f\left(0\right)$
$\therefore k=0$