Properties of Limit at Infinity of Real Function/Multiple Rule

Theorem

Let $a, \alpha \in \R$.

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

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

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

Then:

$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x}$ exists

with:

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

Proof

If $\alpha = 0$, the result follows from Limit of Constant Function: Limit at $\infty$.

Otherwise, since:

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

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

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

Then:

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

Since $\epsilon$ was arbitrary, we have:

$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x}$ exists

with:

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

$\blacksquare$