Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity

Theorem
Let $a \in \R$.

Let $f : \hointr a \infty \to \R$ be a real function such that:


 * $\ds \lim_{x \mathop \to \infty} \map f x$ exists.

Then:


 * $\ds \lim_{x \mathop \to \infty} \map f x = \lim_{x \mathop \to -\infty} \map f {-x}$

where:


 * $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$
 * $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.

Proof
Suppose that:


 * $\ds \lim_{x \mathop \to \infty} \map f x = L$

Then given $\epsilon > 0$ we can find real number $M \ge 0$ such that:


 * $\size {\map f x - L} < \epsilon$ for $x \ge M$.

Then:


 * $\size {\map f {-x} - L} < \epsilon$ for $x \le -M$.

Since $\epsilon$ was arbitrary, we have:


 * $\ds \lim_{x \mathop \to -\infty} \map f {-x} = L$

which was the demand.