# Limit by epsilon-delta proof: Example 2

This is the overdelayed continuation of the discussion on the $\epsilon-\delta$ definition of limits. In this post, we discuss another example.

Prove that the $\lim_{x \to 2} x^2 = 4$.

Recall that the definition states that the limit of $f(x) = L$ as $x$ approaches $a$ if for all $\epsilon > 0$, however small, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

From the example 1, we have learned that we should manipulate $|f(x)-L=|x^2 - 4|$, to make one of the expressions look like $|x-a|=|x-2|$. Solving,  we have

$|f(x) - L| = |x^2 - 4| = |(x+2)(x-2)| = |x+2||x-2|$.

Note that we have accomplished our goal, going back to the definition, this means that if $0 < x - 2 < \delta$, then $|x+2||x-2| < \epsilon$.

Now, it is not possible to divide both sides by $x + 2$ (making it $|x-2| < \frac{\epsilon}{|x+2|})$ because $x$ varies. This means that we have to find a constant $k$ such that $|x + 2| < k$.

If $x$ is confined to some interval centered at $2$, then we can find $k$. For instance, suppose $|x-2| < 1$, which is the same as $1 < x < 3$, then $3 < x+2 < 5$. In particular $|x + 2| = x + 2 < 5$.

Therefore,

$|x+2||x-2| < 5|x-2| <\epsilon$. Hence, we have $|x - 2| < \frac{\epsilon}{5}$. But we have two restrictions:

$|x - 2 | < 1$ and $|x-2| < \frac{\epsilon}{5}$

so, to be sure that both inequalities are obeyed, we choose $\delta$ to be smaller than  $1$ and $\frac{\epsilon}{5}$ abbreviated as  $\delta \leq \min (1, \frac{\epsilon}{5})$ whenever $|x^2 - 4| < \epsilon$.

## 2 thoughts on “Limit by epsilon-delta proof: Example 2”

1. Hi Guillermo Bautista.I’m a 10th grader,and actually practicing Calculus,thus I’m working on Integration by Partial Fractions,doing great.But I felt like I needed to go back a little on Limits and Continuity(though I’m good at it) for the intriguing epsilon-delta definition.I was just wondering if you had some good examples,explanations and proofs about how to deal with non-linear functions(let it be polynomial,logarithmic,exponential,rational and trigonometric functions).Here is my e-mail:smashboy7@gmail.com.I appreciate your help.