If $\lim _{x \rightarrow 3} \left( \frac{x^n - 3^n}{x - 3} \right) = 108$ and $n \in N$,then the value of $n$ is

If $\lim _{x \rightarrow 0} \frac{\alpha x e^{x}-\beta \log _{e}(1+x)+\gamma x^{2} e^{-x}}{x \sin ^{2} x}=10$,where $\alpha, \beta, \gamma \in R$,then the value of $\alpha+\beta+\gamma$ is:

Given $f(x) = \frac{ax + b}{x + 1}$,$\lim_{x \rightarrow \infty} f(x) = 1$ and $\lim_{x \rightarrow 0} f(x) = 2$,then $f(-2)$ is

Let $\alpha, \beta \in R$ be such that $\lim _{x \rightarrow 0} \frac{x^2 \sin (\beta x)}{\alpha x-\sin x}=1$. Then $6(\alpha+\beta)$ equals

If $f: R \to [0, \infty)$ is such that $\lim_{x \to 5} f(x)$ exists and $\lim_{x \to 5} \frac{(f(x))^2 - 9}{\sqrt{|x - 5|}} = 0$,then $\lim_{x \to 5} f(x)$ equals:

Let $f : R - \{0\} \rightarrow R$ be a function such that $f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}$. If $\lim_{x \rightarrow 0} \left(\frac{1}{\alpha x} + f(x)\right) = \beta$,where $\alpha, \beta \in R$,then $\alpha + 2\beta$ is equal to: