Question

If $f\left(x\right)$ is continuous at $x=0$, where
$\begin{array}{c}f\left(x\right)=\frac{1-\cos kx}{x^2},forx\ne 0\\ =\frac{1}{2},forx=0\end{array}$
Find k.

Answer 1

It should be given continuous at $x=0$ because to find k need continuity at $x=0$.It is useless of giving at &&x=10&&.If it is given that it is continuous at $x=0$ then;

we use continuity condition at $x=a$ for a function $f\left(x\right)$, that is

$f(a-)=f(a+)=f\left(a\right)$

from data given,

$f\left(0\right)=\frac{1}{2}$

$f(0-)=f(0+)=\frac{1-\cos kx}{x^2}$

from L'hospital (L'hospital) at $x=0-,x=0+$

$f\left(x\right)=\frac{k\sin kx}{2\times x}$

again applying it;

$f\left(x\right)=\frac{k^2\cos kx}{2}$

putting $x=0$;

$f\left(0\right)=\frac{k^2}{2}$

applying continuity condition gives

$\frac{k^2}{2}=\frac{1}{2}$

$k^2=1$

$k=1,-1$