# How does one prove this statement or provide a counterexample?

## Is it possible to prove the following statement or provide a counterexample function for the following statement? Thanks in advance. ${\lim}_{x \to - \infty} \left(f \left(x\right)\right) = {\lim}_{x \to \infty} \left(f \left(- x\right)\right)$

Sep 7, 2017

It's true.

#### Explanation:

Technically, there are cases to be considered when $f \rightarrow \infty$ and when $f \rightarrow - \infty$. I will illustrate a proof for the case when the limit on the left exists.
Assume ${\lim}_{x \rightarrow - \infty} f \left(x\right) = L$.
Let $\epsilon > 0$.
Since the limit is L, then there exists an M such that whenever $x < M$, we have $| f \left(x\right) - L | < \epsilon$.

Now examine $f \left(- x\right)$. We will introduce a dummy variable, y.
Whenever $y > - M$, it is true that $- y < M$.
For such y, given the above information,
$| f \left(- y\right) - L | < \epsilon$.

Let $N = - M$.
Then there exists an N such that when $y > N$, we have
$| f \left(- y\right) - L | < \epsilon$.
This is the definition of the limit on the right side.