Question

Find the values of k so that the function f is continuous at the indicated point.
f(x)=\left\\{\begin{matrix} kx^2, &if\,x\leq2 \\\ 3,&if\,x>2 \end{matrix}\right. \,at\,x=2

Answer 1

The given function is

f(x)=\left\\{\begin{matrix} kx^2, &if\,x\leq2 \\\ 3,&if\,x>2 \end{matrix}\right.

The given function f is continuous at x=2 if f is defined at x=2, and if the value of the f at x=2 equals the limit of f at x=2.
It is evident that f is defined at x=2 and f(2)=k(2)2=4k
$\lim_{x\rightarrow 2^-}$ f(x)=$\lim_{x\rightarrow 2^+}$f(x)=f(2)
\Rightarrow$$\lim_{x\rightarrow 2^-}(kx2)=$\lim_{x\rightarrow 2^+}$(3)=4k
$\Rightarrow$k$\times$22=3=4k
$\Rightarrow$4k=3=4k
$\Rightarrow$4k=3
k=$\frac{3}{4}$
Therefore, the required value of k is $\frac{3}{4}$.