Question

assume \\( \lim_{x \to 5} f(x) = 26 \\), \\( \lim_{x \to 5} g(x) = 8 \\), and \\( \lim_{x \to 5} h(x) = 6 \\). compute the following limit and state the limit laws used to justify the computations.\\( \lim_{x \to 5} \frac{f(x)}{g(x) - h(x)} \\)\\( \lim_{x \to 5} \frac{f(x)}{g(x) - h(x)} = 13 \\) (simplify your answer.)state the limit law(s) used to justify the computation. select all that apply\\( \square \\) a. root\\( \square \\) b. difference\\( \square \\) c. power\\( \square \\) d. sum\\( \square \\) e. quotient\\( \square \\) f. constant multiple\\( \square \\) g. product

Explanation:

Step1: Apply Quotient Law

The Quotient Law for limits 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 $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$. Here, we have $\lim_{x \to 5} \frac{f(x)}{g(x)-h(x)}$, so first, we need to find $\lim_{x \to 5} (g(x)-h(x))$ and then apply the Quotient Law.

Step2: Apply Difference Law

The Difference Law for limits states that $\lim_{x \to a} (g(x)-h(x)) = \lim_{x \to a} g(x) - \lim_{x \to a} h(x)$, provided both limits exist. Given $\lim_{x \to 5} g(x) = 8$ and $\lim_{x \to 5} h(x) = 6$, we calculate $\lim_{x \to 5} (g(x)-h(x)) = 8 - 6 = 2$.

Step3: Apply Quotient Law

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

Answer:

The limit laws used are B (Difference) and E (Quotient). So the correct options are:
B. Difference
E. Quotient