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

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:

If $\lim _{x \rightarrow 2} \frac{3 x^2-a x+5 b}{x-2}=17$,then $a b=$

Define $f(x) = \begin{cases} b - ax & \text{if } x < 2 \\ 3 & \text{if } x = 2 \\ a + 2bx & \text{if } x > 2 \end{cases}$. If $\lim_{x \rightarrow 2} f(x)$ exists,then find the value of $\frac{a}{b}$.

If $\mathop {\lim }\limits_{x \to 0} \left( {\frac{{3\sin x - 3x + \frac{{{x^3}}}{2}}}{{2{x^n}}}} \right)$ is a finite number,then the greatest value of $n \in N$ is -

Let $k \in \mathbb{R}$. If $\lim _{x \rightarrow 0^{+}}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}= e ^6$,then the value of $k$ is