Question: Prove that $\lim_{x\to\infty}f(x) = \lim_{x\to-\infty}f(-x)$
My Work:
If $\lim_{x\to\infty}f(x) = l$: by definition: $\forall \varepsilon > 0, \exists N$ such that if $x > N$, then $\vert f(x) - l \vert < \varepsilon$
Substitute $-x$ for $x$ that is $-x > N$, then $\vert f(-x) - l \vert < \varepsilon$
but $-x > N$ does not imply $x < N$ which is needed to satisfy the condition and show $\lim_{x\to-\infty}f(-x)$
What am I overlooking?
Thanks