If $\lim _{x \rightarrow 5} \frac{x^{k}-5^{k}}{x-5}=500$,then the value of $k$,where $k \in N$ is

The integral value of $n$ for which $\lim _{x \rightarrow 0} \frac{(\cos x-1)(\cos x-e^x)}{x^n}$ is a finite non-zero real number is

Evaluate the value of $k$ if $\lim _{x \rightarrow 0} \frac{(7^x-1)^4}{\tan (\frac{x}{k}) \cdot \log (1+\frac{x^2}{3}) \cdot \sin 4 x} = 3(\log 7)^3$.

If $\mathop {\lim }\limits_{x \to 0} \frac{{[(a - n)nx - \tan x]\sin nx}}{{{x^2}}} = 0,$ where $n$ is a non-zero real number,then $a$ is equal to

If $\mathop {\lim }\limits_{x \to 2} \frac{{\tan (x - 2)({x^2} + (a - 2)x - 2a)}}{{({x^2} - 4x + 4)}} = 7$,then $a$ is -

If $f(x) = \begin{cases} 3ax - 2b, & x > 1 \\ ax + b + 1, & x < 1 \end{cases}$ and $\lim_{x \rightarrow 1} f(x)$ exists,then the relation between $a$ and $b$ is