Chapter 1.4, Problem 33ES

Use Definition $\mathrm{1.4.1}$ to prove that the stated limit is correct. In each case, to show that ${\lim }_{x\to a}f\left(x\right)=L,$ factor $\left|f\left(x\right)-L\right|$ in the form $\left|f\left(x\right)-L\right|\cdot \left|\text{"something"}\right|\cdot \left|x-a\right|$ and then bound the size of $\left|\text{"something"}\right|$ by putting restrictions on the size of $\delta$ .

$\underset{x\to -2}{\lim }\frac{1}{x+1}=-1$