# Chapter 3 - The Derivative - 3.1 Limits - 3.1 Exercises - Page 137: 28

100

#### Work Step by Step

see: Rules for Limits Let $a, A$, and $B$ be real numbers, and let $f$ and $g$ be functions such that $\displaystyle \lim_{x\rightarrow a}f(x)=A$ and $\displaystyle \lim_{x\rightarrow a}g(x)=B$. 1. If $k$ is a constant, then $\displaystyle \lim_{x\rightarrow a}k=k$ and $\displaystyle \lim_{x\rightarrow a}[k\cdot f(x)]=k\cdot\lim_{x\rightarrow a}f(x)=k\cdot A$. 2. $\displaystyle \lim_{x\rightarrow a}[f(x)\pm g(x)]=\lim_{x\rightarrow a}f(x)\pm\lim_{x\rightarrow a}g(x)=A\pm B$ 6. For any real number $k,$ $\displaystyle \lim_{x\rightarrow a}[f(x)]^{k}=[\lim_{x\rightarrow a}f(x)]^{k}=A^{k}$, provided this limit exists. --------------- $\displaystyle \lim_{x\rightarrow 4}[1+f(x)]^{2}=\qquad$... use rule 6 =$[\displaystyle \lim_{x\rightarrow 4}(1+f(x))]^{2}\qquad$... use rule $2$ $=[\displaystyle \lim_{x\rightarrow 4}1+\lim_{x\rightarrow 4}f(x)]^{2}\qquad$... use rule $1$ $=(1+9)^{2}=10^{2}$ $=100$

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