Question

suppose the function g statisfies 24x - 2x^2 - 68 < g(x) < x^2 - 12x + 40. we want to use the squeeze theorem to evaluate lim(x→6) g(x). first evaluate: lim(x→6) 24x - 2x^2 - 68. next evaluate: lim(x→6) x^2 - 12x + 40. therefore, by the squeeze theorem, lim(x→6) g(x) =

Explanation:

Step1: Evaluate $\lim_{x

ightarrow6}(24x - 2x^{2}-68)$
Substitute $x = 6$ into $24x - 2x^{2}-68$.
$24\times6-2\times6^{2}-68=144 - 2\times36-68=144 - 72 - 68 = 4$

Step2: Evaluate $\lim_{x

ightarrow6}(x^{2}-12x + 40)$
Substitute $x = 6$ into $x^{2}-12x + 40$.
$6^{2}-12\times6 + 40=36-72 + 40 = 4$

Step3: Apply Squeeze Theorem

Since $24x - 2x^{2}-68ightarrow6}(24x - 2x^{2}-68)=\lim_{x
ightarrow6}(x^{2}-12x + 40)=4$, by the Squeeze Theorem, $\lim_{x
ightarrow6}g(x)=4$.

Answer:

$4$