If $\lim_{x \to 3} \frac{x^2 - ax - 3b}{x - 3} = 5$, then $a + b =$

If $\lim _{x \rightarrow 0} \frac{e^{a x}-\cos (b x)-\frac{c x e^{-c x}}{2}}{1-\cos (2 x)}=17$,then $5 a^2+b^2$ is equal to

Let $\tan (2\pi |\sin \theta |) = \cot (2\pi |\cos \theta |)$,where $\theta \in R$ and $f(x) = (\sin^2 \theta + \cos^2 \theta)$. The value of $\lim_{x \to \infty} [\frac{2}{f(x)}]$ equals (Here $[\,]$ represents the greatest integer function).

The largest value of the non-negative integer $a$ for which $\lim _{x \rightarrow 1}\left\{\frac{-a x+\sin (x-1)+a}{x+\sin (x-1)-1}\right\}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4}$ is

$\lim _{x}$ ${\rightarrow -a} \frac{x^7+a^7}{x+a} = 7$ $\Rightarrow a = ?$

If $\lim _{x \rightarrow 1} \frac{x^2-ax+b}{x-1}=7$,then $a+b$ is equal to