Determine the value of the constant k so that the function
Question:

Determine the value of the constant k so that the function

$f(x)=\left\{\begin{array}{ll}k x^{2}, & \text { if } x \leq 2 \\ 3, & \text { if } x>2\end{array}\right.$ is continuous at $x=2$

Solution:

Given:

$f(x)=\left\{\begin{array}{l}k x^{2}, \text { if } x \leq 2 \\ 3, \text { if } x>2\end{array}\right.$

If $f(x)$ is continuous at $x=2$, then

$\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=f(2)$              …..(1)

Now,

$\lim _{x \rightarrow 2^{-}} f(x)=\lim _{h \rightarrow 0} f(2-h)=\lim _{h \rightarrow 0} k(2-h)^{2}=4 k$

And, $f(2)=3$

From (1), we have

$4 k=3$

$\Rightarrow k=\frac{3}{4}$