$\lim _{x \rightarrow 0} \left( \frac{3^{x}-1}{x} \right)$ is equal to

The value of $\lim _{x \rightarrow 1} \frac{x^4-\sqrt{x}}{\sqrt{x}-1}$ is

Let $f(1) = g(1) = k$ and their $n^{th}$ derivatives $f^{(n)}(1), g^{(n)}(1)$ exist and are not equal for some $n$. If $\lim _{x \rightarrow 1} \frac{f(1) g(x) - f(1) - g(1) f(x) + g(1)}{g(x) - f(x)} = 4$,then the value of $k$ is:

Let $f(x) = x^{6} + 2x^{4} + x^{3} + 2x + 3$,$x \in R$. Then the natural number $n$ for which $\lim_{x \rightarrow 1} \frac{x^{n} f(1) - f(x)}{x - 1} = 44$ is ...... .

If $f: R \rightarrow R$ is such that $f(3)=16$ and $f^{\prime}(3)=4$,then find the value of $\lim _{x \rightarrow 3} \frac{x f(3)-3 f(x)}{x-3}$.

The value of $\mathop {\lim }\limits_{x \to a} \frac{{\log (x - a)}}{{\log ({e^x} - {e^a})}}$ is