Question

step 1
note that f is continuous on (-∞, 4) and (4, ∞). for the function to be continuous on (-∞,∞), we need to ensure that as x approaches 4, the left and right limits match.
first we find the left limit.
lim_{x→4^{-}} f(x)=lim_{x→4^{-}} (cx^{2}+9x)=
next, we find the right limit.
lim_{x→4^{+}} f(x)=lim_{x→4^{+}} (x^{3}-cx)=

Explanation:

Step1: Calculate left - hand limit

Substitute \(x = 4\) into \(cx^{2}+9x\).
\(\lim_{x
ightarrow4^{-}}(cx^{2}+9x)=c\times4^{2}+9\times4=16c + 36\)

Step2: Calculate right - hand limit

Substitute \(x = 4\) into \(x^{3}-cx\).
\(\lim_{x
ightarrow4^{+}}(x^{3}-cx)=4^{3}-c\times4=64 - 4c\)

Answer:

Left - hand limit: \(16c + 36\), Right - hand limit: \(64 - 4c\)