Proving that $\begin{equation*} \lim_{x \rightarrow 1} |x + 2|=3 \end{equation*}$ using limit definition.

Question

Prove that $\begin{equation*} \lim_{x \rightarrow 1} |x + 2|=3 \end{equation*}$ using limit definition>

I have a problem when I substitute for $|x + 2|$ by $-x-2$, could anyone help me in this case please? I could not reach $|x-1|$ which is less than $\delta$ in this case.

Answer 1

The crucial trick is that: $\bigg||x+2|-3\bigg|\leq|(x+2)-3|=|x-1|$ by triangle inequality.

@Michael Hardy has demonstrated that triangle inequality is not needed by looking locally at $x=1$, personally I think one should choose $\delta=\min\{1,\epsilon\}$ to assure that $x$ does not vary too far away to get the positiveness of $x+2$.

Answer 2

When $x$ is near $1,$ then $x+2$ is positive, so $|x+2|$ would simplify to just $x+2,$ not $-x-2.$

So if $|x-1|<\delta$ then $|f(x) - 3| = |(x+2) - 3| = |x-1|<\delta.$ Thus setting $\delta=\varepsilon$ suffices.