Question

For what value(s) of k, is the function f defined by
f(x) = {sin (1 + x) - sin(1- x) x not equal to 0
{k cos 1, x= 0
Continuous at x = 0 ?

Answer 1

The given function

$$f(x)=\begin{cases}\sin(1+x)-\sin(1-x)&x\neq0\\k\cos1&x=0\end{cases}$$

is continuous, if

$$\lim_{x \to 0}(\sin(1+x)-\sin(1-x))=\lim_{x \to 0}k\cos1$$

Let us solve this equation with respect to k

$$\sin1-\sin1=k\cos1$$ $$k\cos1=0$$ $$k=0$$

Thus, the function f(x) is continuous for k = 0.