Question

assume \\( \lim\limits_{x \to 5} f(x) = 26 \\), \\( \lim\limits_{x \to 5} g(x) = 8 \\), and \\( \lim\limits_{x \to 5} h(x) = 6 \\). compute the following limit and state the limit laws used to justify the computations.\\( \lim\limits_{x \to 5} \frac{f(x)}{g(x) - h(x)} \\)\\( \lim\limits_{x \to 5} \frac{f(x)}{g(x) - h(x)} = \square \\)\\( \text{(simplify your answer.)} \\)

Explanation:

Step1: Apply Quotient Law

The Quotient Law states that if $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist and $\lim_{x \to a} g(x)
eq 0$, then $\lim_{x \to a} \frac{f(x)}{g(x)}=\frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$. Also, the Difference Law states that $\lim_{x \to a} (g(x)-h(x))=\lim_{x \to a} g(x)-\lim_{x \to a} h(x)$. First, we find the limit of the denominator $g(x)-h(x)$ as $x \to 5$.
Using the Difference Law: $\lim_{x \to 5}(g(x)-h(x))=\lim_{x \to 5}g(x)-\lim_{x \to 5}h(x)=8 - 6=2$.

Step2: Apply Quotient Law to the whole expression

Now, for the limit $\lim_{x \to 5}\frac{f(x)}{g(x)-h(x)}$, using the Quotient Law (since $\lim_{x \to 5}f(x) = 26$ and $\lim_{x \to 5}(g(x)-h(x))=2
eq0$), we have $\lim_{x \to 5}\frac{f(x)}{g(x)-h(x)}=\frac{\lim_{x \to 5}f(x)}{\lim_{x \to 5}(g(x)-h(x))}=\frac{26}{2}=13$.

Answer:

13