If $f(a)=2, f^{\prime}(a)=1, g(a)=-1, g^{\prime}(a)=2$,then as $x$ approaches $a$,the limit of $\frac{g(x) f(a)-g(a) f(x)}{x-a}$ is

Let $f(x)$ be a differentiable function and $f^{\prime}(4)=5$. Then,$\lim _{x \rightarrow 2} \frac{f(4) - f\left(x^{2}\right)}{x-2}$ equals

$\lim _{x \rightarrow \frac{\pi}{2}} \left(\frac{2x-\pi}{\cos x}\right)$ is equal to

The limit of $\left\{\frac{1}{x} \sqrt{1+x}-\sqrt{1+\frac{1}{x^{2}}}\right\}$ as $x \rightarrow 0$ is:

If $\alpha, \beta$ are the distinct roots of $x^{2}+bx+c=0$,then $\lim _{x \rightarrow \beta} \frac{e^{2(x^{2}+bx+c)}-1-2(x^{2}+bx+c)}{(x-\beta)^{2}}$ is equal to:

$\mathop {\lim }\limits_{x \to 0} \frac{{\cos ax - \cos bx}}{{{x^2}}} = $