Question

Find the values of $k$ so that the function $f$ is continuous at the indicated point:
$f\left(x\right)=$ $\{\begin{array}{c}k{x}^2,ifx\le 2\\ 3,ifx>2\end{array}$ at $x=2$

Answer 1

The given function is $f\left(x\right)=\{\begin{array}{c}k{x}^2,\text{if}x\le 2\\ 3,\text{if}x>2\end{array}$
The given function $f$ is continuous at $x=2$
$\underset{x\to 2}{lim}f\left(x\right)=\underset{x\to 2}{lim}f\left(x\right)=f\left(2\right)$
$\Rightarrow \underset{x\to 2}{lim}\left(k{x}^2\right)=\underset{x\to 2}{lim}\left(3\right)=4k$
$\Rightarrow k\cdot 2^2=3=4k$
$\Rightarrow 4k=3=4k$

$\Rightarrow 4k=3$

$\Rightarrow k=\frac{3}{4}$
Therefore, the required value of $k$ is $\frac{3}{4}$.