Question

assume \\(\lim_{x \to 5} f(x) = 14\\), \\(\lim_{x \to 5} g(x) = 7\\), and \\(\lim_{x \to 5} h(x) = 5\\). compute the following limit and state the limit laws used to justify the computation.

\\\lim_{x \to 5} \frac{f(x)}{g(x) - h(x)}\\

select each limit law used to justify the computation.

a. difference
b. product
c. sum
d. power
e. constant multiple
f. root
g. quotient

Explanation:

Evaluate the limit expression

We want to compute the limit:

$$ L = \lim_{x \to 5} \frac{f(x)}{g(x) - h(x)} $$

Given:

$$ \lim_{x \to 5} f(x) = 14, \quad \lim_{x \to 5} g(x) = 7, \quad \lim_{x \to 5} h(x) = 5 $$

Apply the Quotient Law

Using the Quotient Law for Limits, we can split the limit of the quotient into the quotient of the limits, provided the limit of the denominator is non-zero:

$$ \lim_{x \to 5} \frac{f(x)}{g(x) - h(x)} = \frac{\lim_{x \to 5} f(x)}{\lim_{x \to 5} (g(x) - h(x))} $$

Apply the Difference Law

Using the Difference Law for Limits, we split the limit in the denominator:

$$ \lim_{x \to 5} (g(x) - h(x)) = \lim_{x \to 5} g(x) - \lim_{x \to 5} h(x) $$

Calculate the final numerical value

Substitute the given limit values into our expression:

$$ L = \frac{14}{7 - 5} = \frac{14}{2} = 7 $$

Since the denominator \(7 - 5 = 2
eq 0\), applying the Limit Laws is fully justified.

Identify the selected laws

The laws used to justify this computation are:

• Quotient Law for Limits (since we took the limit of a quotient)
• Difference Law for Limits (since we took the limit of a difference in the denominator)

Answer:

Question 1

The value of the limit is:

$$ 7 $$

Question 2

The limit laws used to justify the computation are:

• A. Difference
• G. Quotient