If $f(x) = \frac{10^x + 7^x - 14^x - 5^x}{1 - \cos x}$ for $x \neq 0$ is continuous at $x = 0$,then the value of $f(0)$ is:

The value of $k$,for which the function $f(x) = \begin{cases} (\frac{4}{5})^{\frac{\tan 4x}{\tan 5x}}, & 0 < x < \frac{\pi}{2} \\ k + \frac{2}{5}, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$,is:

Let $f: R \rightarrow R$ be defined by $f(x)=\begin{cases} \alpha+\frac{\sin [x]}{x}, & x>0 \\ 2, & x=0 \\ \beta+\left[\frac{\sin x-x}{x^3}\right], & x < 0 \end{cases}$. If $f$ is continuous at $x=0$,find the value of $\alpha + \beta$.

Let $f: R \rightarrow R$ be defined as $f(x) = \begin{cases} \frac{x^{3}}{(1-\cos 2x)^{2}} \log_{e}\left(\frac{1+2xe^{-2x}}{(1-xe^{-x})^{2}}\right), & x \neq 0 \\ \alpha, & x=0 \end{cases}$. If $f$ is continuous at $x=0$,then $\alpha$ is equal to:

If the function $f(x) = \begin{cases} \frac{\cos ax - \cos bx}{\cos cx - \cos bx} & , x \neq 0 \\ -1 & , x = 0 \end{cases}$ is continuous at $x = 0$,then $a^2, b^2, c^2$ are in

If $f(x) = \begin{cases} \frac{x-2}{|x-2|}+a & , x<2 \\ a+b & , x=2 \\ \frac{x-2}{|x-2|}+b & , x>2 \end{cases}$ is continuous at $x=2$, then $a+b=$