In this post, I show that

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

given that $\lim_{x \rightarrow a} f(x) = A$ and $\lim_{x \rightarrow a} g(x) = B$. To do this, I approximately follow the steps in Reference [1].

Objective:

The objective is to directly show that

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

Proof:

$\lim_{x\rightarrow a}[f(x) – g(x)]$

From arithmetic,

$=\lim_{x\rightarrow a}[f(x) + (-1)g(x)]$

From this property,

$=\lim_{x\rightarrow a}f(x) + \lim_{x\rightarrow a}(-1)g(x)$

From this property,

$=\lim_{x\rightarrow a}f(x) + (-1)\lim_{x\rightarrow a}g(x)$

From arithmetic,

$=\lim_{x\rightarrow a}f(x) – \lim_{x\rightarrow a}g(x)$

$\square$

References: