Question

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->-oo)(f(x))=lim_(x->oo)(f(-x))`

Answer 1

It's true.

Explanation:

Technically, there are cases to be considered when `f rarr oo` and when `f rarr -oo`. I will illustrate a proof for the case when the limit on the left exists.
Assume `lim_(x rarr -oo) f(x) = L`.
Let `epsilon > 0`.
Since the limit is L, then there exists an M such that whenever `x < M`, we have `|f(x) - L| < epsilon`.

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

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