Proving an Incorrect Limit of a Function

# Proving an Incorrect Limit of a Function

Recall that if we write $\lim_{x \to a} f(x) = L$, then this means that as $x \to a$, $f(x) \to L$, or rather, $\forall \epsilon > 0 \: \exists \delta > 0 \: \mathrm{s.t.} \: \forall x : 0 < \mid x - a \mid < \delta, \: \mid f(x) - L \mid < \epsilon$. However, sometimes we don't necessarily know for sure if $\lim_{x \to a} f(x) = L$.

For example, consider the statement $\lim_{x \to 1} x = 2$. We know for a fact that this is untrue, though, suppose that we don't and that we want to prove this limit is incorrect, or rather, prove that $\lim_{x \to 1} x ≠ 2$. We will essentially do this by showing that $\exists \epsilon > 0 \: \forall \delta > 0 \: \mathrm{s.t.} \exists x : 0 < \mid x - a \mid < \delta, \: \mid f(x) - L \mid ≥ \epsilon$.

 Note: Notice the difference between what $lim_{x \to a} f(x) = L$ implies and $\lim_{x \to a} f(x) ≠ L \Leftrightarrow \exists \epsilon > 0 \: \forall \delta > 0 \: \mathrm{s.t.} \exists x : 0 < \mid x - a \mid < \delta, \: \mid f(x) - L \mid ≥ \epsilon$. To show that a limit is not $L$, we have essentially negated the definition of a limit existing as $L$.

Now let's look at an example applying the negation of the definition of a limit.

## Example 1

Prove that $\lim_{x \to 1} x ≠ 2$.