Question

question 19
suppose the function g statisfies
$16x - 2x^2 - 24 < g(x) < x^2 - 8x + 24$
we want to use the squeeze theorem to evaluate $lim_{x \to 4} g(x)$
first evaluate: $lim_{x \to 4} 16x - 2x^2 - 24$
next evaluate: $lim_{x \to 4} x^2 - 8x + 24$
therefore, by the squeeze theorem, $lim_{x \to 4} g(x) = $

Explanation:

Step1: Substitute $x=4$ into first function

$\lim_{x \to 4} (16x - 2x^2 - 24) = 16(4) - 2(4)^2 - 24$

Step2: Calculate first limit

$= 64 - 32 - 24 = 8$

Step3: Substitute $x=4$ into second function

$\lim_{x \to 4} (x^2 - 8x + 24) = (4)^2 - 8(4) + 24$

Step4: Calculate second limit

$= 16 - 32 + 24 = 8$

Step5: Apply Squeeze Theorem

Since $8 < \lim_{x \to 4}g(x) < 8$, the limit equals 8.

Answer:

$\lim_{x \to 4} (16x - 2x^2 - 24) = 8$
$\lim_{x \to 4} (x^2 - 8x + 24) = 8$
$\lim_{x \to 4} g(x) = 8$