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 if and only if $\ds \lim_{x \mathop \to \infty} \map f {-x}$ 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:

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

if and only if 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 if and only if $\ds \lim_{x \mathop \to \infty} \map f {-x}$ exists

with:

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

which was the demand.

$\blacksquare$