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)}\\

\\(\lim_{x \to 5} \frac{f(x)}{g(x) - h(x)} = \\) (simplify your answer.)

Explanation:

Apply the Quotient Law for Limits

Using the Limit Laws knowledge point

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

Apply the Difference Law for Limits

Using the Limit Laws knowledge point

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

Substitute the given limit values

Using the Limit Laws knowledge point

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

Simplify the numerical expression

Using the Limit Laws knowledge point

$$ \frac{14}{2} = 7 $$

Answer:

Assume \(\lim_{x \to 5} f(x) = 14\), \(\lim_{x \to 5} g(x) = 7\), and \(\lim_{x \to 5} h(x) = 5\).

\(\lim_{x \to 5} \frac{f(x)}{g(x) - h(x)} =\) <blank>7</blank>