Question

find the value of the constant c that makes the following function continuous.
f(x)=\begin{cases}cx + 6&\text{if }xin(-infty,8\cx^{2}-6&\text{if }xin(8,infty)end{cases}
c =

Explanation:

Step1: Recall continuity condition

For a function to be continuous at $x = 8$, $\lim_{x
ightarrow8^{-}}f(x)=\lim_{x
ightarrow8^{+}}f(x)$.
$\lim_{x
ightarrow8^{-}}(cx + 6)=c\times8+6 = 8c+6$.
$\lim_{x
ightarrow8^{+}}(cx^{2}-6)=c\times8^{2}-6=64c - 6$.

Step2: Set the left - hand and right - hand limits equal

Set $8c + 6=64c-6$.
First, move the terms with $c$ to one side: $6 + 6=64c-8c$.
Then simplify: $12 = 56c$.
Solve for $c$: $c=\frac{12}{56}=\frac{3}{14}$.

Answer:

$\frac{3}{14}$