## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 11 - Section 11.2 - Finding Limits Using Properties of Limits - Concept and Vocabulary Check - Page 1153: 11

#### Answer

The equation “$\underset{x\to 6}{\mathop{\lim }}\,\text{ }\frac{{{x}^{2}}-25}{x-5}=\frac{\underset{x\to 6}{\mathop{\lim }}\,\text{ }\left( {{x}^{2}}-25 \right)}{\underset{x\to 6}{\mathop{\lim }}\,\text{ }\left( x-5 \right)}$” is true.

#### Work Step by Step

The limit of the quotient of two functions equals the quotient of their limits when the limit of the denominator is not zero. That is: The limit of a quotient: If $\underset{x\to a}{\mathop{\lim }}\,\text{ }f\left( x \right)=L\text{ and }\underset{x\to a}{\mathop{\lim }}\,\text{ }g\left( x \right)=M\text{ };M\ne 0$, then $\underset{x\to a}{\mathop{\lim }}\,\text{ }\frac{f\left( x \right)}{g\left( x \right)}=\frac{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}{\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)}=\frac{L}{M}\text{ };M\ne 0$ This is the quotient property. Before applying the quotient property, find the limit of the denominator. If this limit is not zero, apply the quotient property. And if the limit of the denominator is zero, the quotient property cannot be used. Now, the two functions in this limit problem are $f\left( x \right)={{x}^{2}}-25$ and $g\left( x \right)=x-5$. The limit of the quotient of these functions is $6$. This means $x$ is approaching to 6. The first step is to find the limit of the denominator, $g\left( x \right)$. \begin{align} & \underset{x\to 6}{\mathop{\lim }}\,\text{ }x-5=6-5 \\ & =1 \end{align} Since, the limit of the denominator is not zero, the quotient property can be used. Therefore, the statement “$\underset{x\to 6}{\mathop{\lim }}\,\text{ }\frac{{{x}^{2}}-25}{x-5}=\frac{\underset{x\to 6}{\mathop{\lim }}\,\text{ }\left( {{x}^{2}}-25 \right)}{\underset{x\to 6}{\mathop{\lim }}\,\text{ }\left( x-5 \right)}$” is true.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.