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 \(\frac{3}{4}\).