Question

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;  \[f\left( x \right) = \begin{cases}k x^2 , & x \geq 1 \\ 4 , & x < 1\end{cases}\]at x = 1

 

Answer 1

Given : 

\[f\left( x \right) = \binom{k x^2 , x \geq 1}{4, x < 1}\] 

We have
(LHL at x = 1) =  

\[\lim_{x \to 1^-} f\left( x \right) = \lim_{h \to 0} f\left( 1 - h \right) = \lim_{h \to 0} 4 = 4\]

(RHL at x = 1) =  \[\lim_{x \to 1^+} f\left( x \right) = \lim_{h \to 0} f\left( 1 + h \right) = \lim_{h \to 0} k \left( 1 + h \right)^2 = k\]

If f(x) is continuous at x = 1, then

\[\lim_{x \to 1^-} f\left( x \right) = \lim_{x \to 1^+} f\left( x \right)\]
\[ \Rightarrow k = 4\]