Question

use lhôpitals rule to evaluate \\( \lim _{x \
ightarrow 10} \frac{x - 10}{x^{2} - 100} \\). then determine the limit using limit laws and commonly known limits. use lhôpitals rule to rewrite the given limit so that it is not an indeterminate form. \\( \lim _{x \
ightarrow 10} \frac{x - 10}{x^{2} - 100}=\lim _{x \
ightarrow 10} \\)

Explanation:

Step1: Apply L'Hôpital's Rule

L'Hôpital's Rule states that if \(\lim_{x
ightarrow a}\frac{f(x)}{g(x)}\) is in the indeterminate form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then \(\lim_{x
ightarrow a}\frac{f(x)}{g(x)}=\lim_{x
ightarrow a}\frac{f^{\prime}(x)}{g^{\prime}(x)}\).
For \(f(x)=x - 10\), \(f^{\prime}(x)=1\). For \(g(x)=x^{2}-100\), \(g^{\prime}(x)=2x\).
So \(\lim_{x
ightarrow 10}\frac{x - 10}{x^{2}-100}=\lim_{x
ightarrow 10}\frac{1}{2x}\)

Answer:

\(\lim_{x
ightarrow 10}\frac{x - 10}{x^{2}-100}=\lim_{x
ightarrow 10}\frac{1}{2x}\)