Answer:
[tex]\lim_{x \to 3} (\frac{f(x)}{g(x)} )=- \infty[/tex]
Step-by-step explanation:
Given that:
f(x) approaches 150, and
g(x) approaches 0, with g(x) < 0,
as x approaches 3.
This means:
[tex]\lim_{x \to 3} f(x)=150 \\\\ \lim_{x \to 3} g(x)=0[/tex]
We need to evaluate:
[tex]\lim_{x \to 3} (\frac{f(x)}{g(x)} )[/tex]
Distributing the limit to numerator and denominator, we get:
[tex]\frac{ \lim_{x \to 0} f(x) }{ \lim_{x \to 0} g(x)}\\\\ = \frac{150}{0}[/tex]
The expression will result in infinity as the answer, but since, g(x) < 0, this means g(x) is approaching 0 from the negative side. As a result, the expression 150/0 will approach negative infinity as x will approach 3.
Therefore, we can conclude:
[tex]\lim_{x \to 3} (\frac{f(x)}{g(x)} )=- \infty[/tex]
