In this post, I show that

$\lim_{x \rightarrow a} [c f(x)] = c \lim_{x \rightarrow a} [f(x)]$

if $c$ is a variable for any real number. To do this, I approximately follow the steps in reference .

Essential Background Information:

A function $f(x)$ approaches a limit $A$ as $x$ approaches $a$ if, and only if, for each positive number $\epsilon$ there is another, $\delta$, such that whenever $0 < |x-a| < \delta$ we have $|f(x) – A| < \epsilon$. That is, when $x$ is near $a$ (within a distance $\delta$ from it), $f(x)$ is near $A$ (within a distance $\epsilon$ from it). In symbols we write $\lim_{x \rightarrow a} f(x) = A$.

Objective:

Suppose that $|f(x) – A| < \epsilon$ whenever $0 < |x-a| < \delta$.

For $c \ne 0$, the objective is to show that

$|cf(x) – cA| < \epsilon_1$ whenever $0 < |x-a| < \delta$,

in which $\epsilon_1$ is a positive number.

For $c=0$, the objective is to directly show that

$\lim_{x \rightarrow a} [c f(x)] = c \lim_{x \rightarrow a} [f(x)]$.

Proof:

First consider the case of $c=0$. Suppose that each value of $f$ is a real number. Then

$\lim_{x \rightarrow a} [0 f(x)] = \lim_{x \rightarrow a} $

From this property,

$\lim_{x \rightarrow a}  = 0$.

Assume that $A$ is a variable for any real number. Then

$0 = 0 \lim_{x \rightarrow a} [f(x)]$,

so it has been shown that for $c=0$

$\lim_{x \rightarrow a} [c f(x)] = c \lim_{x \rightarrow a} [f(x)]$.

Now suppose that $c$ is a nonzero real number.

Recall that $|f(x) – A| < \epsilon$ whenever $0 < |x-a| < \delta$. Next multiply $|f(x) – A| < \epsilon$ by $|c|$ to obtain $|c| |f(x) – A| < |c| \epsilon$ whenever $0 < |x-a| < \delta$.

The next step is to use this absolute value property:

$|c| |f(x) – A| = | c (f(x) – A) | = | c f(x) – cA |$.

Hence

$| c f(x) – cA | < |c| \epsilon$ whenever $0 < |x-a| < \delta$.

Next define

$|c| \epsilon \equiv \epsilon_1$. Note that $\epsilon_1 > 0$ as needed since $\epsilon > 0$ and $|c|>0$.

Then, if $c \ne 0$,

$| c f(x) – cA | < \epsilon_1$ whenever $0 < |x-a| < \delta$,

with $\epsilon_1 >0$.

$\square$

References:

 https://tutorial.math.lamar.edu/classes/calci/limitproofs.aspx