Question

In each of the following, find the value of the constant k so that the given function is continuous at the indicated point;  \[f\left( x \right) = \begin{cases}kx + 1, if & x \leq 5 \\ 3x - 5, if & x > 5\end{cases}\] at x = 5

Answer 1

Given : 

\[f\left( x \right) = \binom{kx + 1, \text{ if }  x \leq 5}{3x - 5, \text{ if } x > 5}\]

 We have
(LHL at x = 5) = 

\[\lim_{x \to 5^-} f\left( x \right) = \lim_{h \to 0} f\left( 5 - h \right) = \lim_{h \to 0} k\left( 5 - h \right) + 1 = 5k + 1\]

(RHL at x = 5) = 

\[\lim_{x \to 5^+} f\left( x \right) = \lim_{h \to 0} f\left( 5 + h \right) = \lim_{h \to 0} 3\left( 5 + h \right) - 5 = 10\]

If f(x) is continuous at x = 5, then

\[\lim_{x \to 5^-} f\left( x \right) = \lim_{x \to 5^+} f\left( x \right)\]
\[ \Rightarrow 5k + 1 = 10\]
\[ \Rightarrow k = \frac{9}{5}\]