Questions: Step 1 Recall that a function f is continuous at a number a if the following holds. lim x -> a f(x) = f(a) We are given the following. f(x) = cx^2 + 3x if x < 6 x^3 - cx if x >= 6 We note that for all values of the constant c the function f is continuous on both (-∞, 6) and (6, ∞). The only case that we need to consider is when x=6. To do so, we must determine the value of c such that the following holds. lim x -> 6^- f(x) = lim x -> 6^+ f(x) Find the following limits. lim x -> 6^- f(x) = lim x -> 6^- (cx^2 + 3x) = lim x -> 6^+ f(x) = lim x -> 6^+ (x^3 - cx) =

Transcript text: Step 1 Recall that a function $f$ is continuous at a number a if the following holds. \[ \lim _{x \rightarrow a} f(x)=f(a) \] We are given the following. \[ f(x)=\left\{\begin{array}{ll} c x^{2}+3 x & \text { if } x<6 \\ x^{3}-c x & \text { if } x \geq 6 \end{array}\right. \] We note that for all values of the constant $c$ the function $f$ is continuoles on both $(-\infty, 6)$ and $(6, \infty)$. The only case that we need to consider is whe $x=6$. To do so, we must determine the value of $c$ such that the following holds. \[ \lim _{x \rightarrow 6^{-}} f(x)=\lim _{x \rightarrow 6^{+}} f(x) \] Find the following limits. \[ \begin{aligned} \lim _{x \rightarrow 6^{-}} f(x) & =\lim _{x \rightarrow 6^{-}}\left(c x^{2}+3 x\right) \\ & =\square \end{aligned} \] \[ \begin{aligned} \lim _{x \rightarrow 6^{+}} f(x) & =\lim _{x \rightarrow 6^{+}}\left(x^{3}-c x\right) \\ & =\square \end{aligned} \]