List of top Mathematics Questions on Continuity asked in CBSE Class XII

Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

Show that the function $f(x) = \begin{cases} \frac{\cos x}{\frac{\pi}{2} - x}, & x \neq \frac{\pi}{2} \\ 1, & x = \frac{\pi}{2} \end{cases}$ is continuous at $x = \frac{\pi}{2}$.

The value of \(k\) for which the function \[ f(x)= \begin{cases} x^2\sin\!\left(\dfrac{1}{x}\right), & x\ne0,\\[6pt] k(x+1), & x=0, \end{cases} \] is continuous, is \[ \_\_\_\_. \]

Sports car racing is a form of motorsport which uses sports car prototypes. The competition is held on special tracks designed in various shapes. The piecewise function describing the boundary of one such track is given as follows: \[ f(x) = \begin{cases} x^2 + 4, & 0 \le x \\ 4x^2 - 4x + 40, & x \ge 3 \end{cases} \] Based on the given information, answer the following question: