Question

question 18
suppose the function ( g ) satisfies
( 36x - 2x^2 - 158 < g(x) < x^2 - 18x + 85 ).
we want to use the squeeze theorem to evaluate ( limlimits_{x \to 9} g(x) ).
first evaluate: ( limlimits_{x \to 9} 36x - 2x^2 - 158 )
next evaluate: ( limlimits_{x \to 9} x^2 - 18x + 85 )
therefore, by the squeeze theorem, ( limlimits_{x \to 9} g(x) = )

Explanation:

Step1: Substitute $x=9$ into first function

$\lim_{x \to 9} 36x - 2x^2 - 158 = 36(9) - 2(9)^2 - 158$
$= 324 - 2(81) - 158$
$= 324 - 162 - 158$

Step2: Calculate first limit result

$324 - 162 - 158 = 4$

Step3: Substitute $x=9$ into second function

$\lim_{x \to 9} x^2 - 18x + 85 = (9)^2 - 18(9) + 85$
$= 81 - 162 + 85$

Step4: Calculate second limit result

$81 - 162 + 85 = 4$

Step5: Apply Squeeze Theorem

Since $4 < \lim_{x \to 9} g(x) < 4$, the limit equals 4.

Answer:

$\lim_{x \to 9} 36x - 2x^2 - 158 = 4$
$\lim_{x \to 9} x^2 - 18x + 85 = 4$
$\lim_{x \to 9} g(x) = 4$