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

Theorem
Let $a \in \R$.

Let $f : \hointl {-\infty} a \to \R$ be a real function.

Then:


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

and in this case:


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

where:


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

Proof
Note that:


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

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


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

This is equivalent to:


 * given $\epsilon > 0$ we can find real number $M \le 0$ such that $\size {\map f x - L} < \epsilon$ for $x \ge -M$.

Since $\epsilon$ was arbitrary and $-M \ge 0$, we have:


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

with:


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

which was the demand.