# Chapter 1 - Functions and Limits - 1.8 Continuity - 1.8 Exercises - Page 92: 12

$g$ is continuous at $a=2$

#### Work Step by Step

A function $f$ is continuous at a number $a$ if $\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$ ------------- $\displaystyle \lim_{t\rightarrow 2}g(t)=\lim_{t\rightarrow 2}\frac{t^{2}+5t}{2t+1}=\quad ...$The limit of a quotient $=\displaystyle \frac{\lim_{t\rightarrow 2}(t^{2}+5t)}{\lim_{t\rightarrow 2}(2t+1)}= \quad$...The limit of a sum (twice) $=\displaystyle \frac{\lim_{t\rightarrow 2}t^{2}+\lim_{t\rightarrow 2}5t}{\lim_{t\rightarrow 2}2t+\lim_{t\rightarrow 2}1}\quad$...The limit of a constant times a function $=\displaystyle \frac{\lim_{t\rightarrow 2}t^{2}+5\lim_{t\rightarrow 2}t}{2\lim_{t\rightarrow 2}t+\lim_{t\rightarrow 2}1}\quad$...evaluate $=\displaystyle \frac{2^{2}+5(2)}{2(2)+1}$ $=\displaystyle \frac{14}{5}$ $g(2)=\displaystyle \frac{2^{2}+5(2)}{2(2)+1}=\frac{14}{5}=\lim_{t\rightarrow 2}g(t).$ By the definition, $g$ is continuous at $a=2$.

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