Given,$\sin x = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^{2n-1}}{(2n-1)!}$. If the function $f(x)$ given by $f(x) = \frac{\cos(\sin x) - \cos x}{x^4}$ for $x \neq 0$ and $f(0) = k$ is continuous at $x = 0$,then $k =$

Let $[t]$ represent the greatest integer not exceeding $t$. Then the number of points of discontinuity of $f(x) = [10^x]$ in the interval $(0, 10)$ is:

If $\{x\} = x - [x]$ where $[x]$ is the greatest integer $\leq x$ and $\lim_{x \rightarrow 0^{-}} \frac{\cos^{-1}(1-\{x\}^2) \sin^{-1}(1-\{x\})}{\{x\}-\{x\}^4} = \theta$,then $\tan \theta =$

Which of the following statements is true for the graph of the function $f(x) = \log x$?

The number of points where the function $f(x) = \begin{cases} |2x^2 - 3x - 7| & \text{if } x \leq -1 \\ [4x^2 - 1] & \text{if } -1 < x < 1 \\ |x+1| + |x-2| & \text{if } x \geq 1 \end{cases}$ is discontinuous,where $[t]$ denotes the greatest integer $\leq t$,is:

If $f(x) = \left[\tan \left(\frac{\pi}{4} + x\right)\right]^{\frac{1}{x}}$ for $x \neq 0$ and $f(x) = k$ for $x = 0$ is continuous at $x = 0$,then $k = \dots$