Question

part 1 of 2
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⁻) (cx² + 3x)=
next, we find the right limit.
lim(x→4⁺) f(x)=lim(x→4⁺) (x³ - cx)=

Explanation:

Step1: Calculate left - hand limit

We substitute \(x = 4\) into \(ax^{2}+3x\). So, \(\lim_{x
ightarrow4^{-}}f(x)=\lim_{x
ightarrow4^{-}}(ax^{2}+3x)=a\times4^{2}+3\times4=16a + 12\).

Step2: Calculate right - hand limit

We substitute \(x = 4\) into \(x^{3}-cx\). So, \(\lim_{x
ightarrow4^{+}}f(x)=\lim_{x
ightarrow4^{+}}(x^{3}-cx)=4^{3}-c\times4=64 - 4c\).

Answer:

Left - hand limit: \(16a + 12\); Right - hand limit: \(64 - 4c\)