$\mathop {\lim }\limits_{x \to 0} {(1 - ax)^{\frac{1}{x}}} = $

$\lim _{x \rightarrow 0}(1+3x)^{\frac{2}{x}} = $

The value of $\mathop {\lim }\limits_{x \to 0} \left( \left[ \frac{100x}{\sin x} \right] + \left[ \frac{99 \sin x}{x} \right] \right)$,where $[.]$ denotes the greatest integer function,is

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 1} \frac{4x+3}{x-2}$

If $\lim_{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(nk+1)+(nk+2)+\ldots+(nk+n)] = 33 \cdot \lim_{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot [1^k + 2^k + 3^k + \ldots + n^k]$,then the integral value of $k$ is equal to $....$

$\mathop {\lim }\limits_{x \to a} f(x) \cdot g(x)$ exists,if