Answer
$$\lim_{x \to -2}(4x+5)=-3$$For the proof by using the $\epsilon - \delta$ definition, see below.
Work Step by Step
$$\lim_{x \to -2} (4x+5)=4(-2)+5=-3$$ Now, we want to prove this limit by using $\epsilon - \delta$ definition; that is, we must show that for each $\epsilon >0$, there exists a $\delta >0$ such that $|(4x+5)-(-3)|< \epsilon$ whenever $|x-(-2)|< \delta$.
Now, we have$$|(4x+5)-(-3)|=|(4x+5)+3|=|4(x+2)|=4|x+2|.$$ So, by choosing $\delta =\frac{ \epsilon }{4}$ we conclude that$$|x-(-2)|=|x+2|< \delta = \frac{ \epsilon }{4} \quad \Rightarrow \quad |(4x+5)-(-3)|=4|x+2|$$.
