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: Start with the left side, $\lim_{x\rightarrow a}[f(x) – g(x)]$ 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: [1] https://tutorial.math.lamar.edu/classes/calci/limitproofs.aspx