If $\mathop {\lim }\limits_{x \to 5} \frac{{{x^k} - {5^k}}}{{x - 5}} = 500$,then the positive integral value of $k$ is

Find $\mathop {\lim }\limits_{x \to 0} f(x)$ and $\mathop {\lim }\limits_{x \to 1} f(x),$ where $f(x) = \begin{cases} 2x+3, & x \leq 0 \\ 3(x+1), & x > 0 \end{cases}$

If $\mathop {\lim }\limits_{n \to \infty } \frac{1}{{10 + {{\left( {2\cos x} \right)}^{2n}}}} = 0$,then the complete set of all possible values of $|\sin x|$ is:

If $f(x) = \text{Sgn}(\text{Sgn}(\text{Sgn}(x)))$,then $\mathop {\lim }\limits_{x \to 0} f(x)$ is equal to :-

If $f(x) = \begin{cases} \frac{\sin(1+[x])}{[x]}, & \text{for } [x] \neq 0 \\ 0, & \text{for } [x] = 0 \end{cases}$ where $[x]$ denotes the greatest integer function,then $\lim_{x \rightarrow 0^{-}} f(x)$ is equal to

If $f(x) = \cos x$ when $x = n\pi$ $(n = 0, 1, 2, 3, \dots)$ and $f(x) = 3$ otherwise,and $\phi(x) = \begin{cases} x^2 + 1 & \text{when } x \neq 3, x \neq 0 \\ 3 & \text{when } x = 0 \\ 5 & \text{when } x = 3 \end{cases}$,then find $\lim_{x \to 0} f(\phi(x))$.