$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} \frac{\cos ^{2} x-\sin ^{2} x-1}{\sqrt{x^{2}+1}-1} \cdot \frac{\sqrt{x^{2}+1}+1}{\sqrt{x^{2}+1}+1}$ (Rationalisation) $\lim _{x \rightarrow 0^{+}}-\frac{2 \sin ^{2} x}{x^{2}} \cdot\left(\sqrt{x^{2}+1}+1\right)=-4$
$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{-}} \frac{1}{x} \ell n\left(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}\right)$
$\lim _{x \rightarrow 0^{-}} \frac{\ln \left(1+\frac{x}{a}\right)}{\left(\frac{x}{a}\right) \cdot a}+\frac{\ell n\left(1-\frac{x}{b}\right)}{\left(-\frac{x}{b}\right) \cdot b}$
$=\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{~b}}$
So $\frac{1}{a}+\frac{1}{b}=-4=k$
$\Rightarrow \frac{1}{a}+\frac{1}{b}+\frac{4}{k}=-4-1=-5$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.