Transitivity of the Limit

Question

Let $f$ and $g$ be functions of a single, real variable. Write $f \rightarrow c$ (as $x \rightarrow c'$) for $\lim_{x\rightarrow c'}f(x)= c$. If $f \rightarrow c$ and $g \rightarrow c$ as $x \rightarrow c'$, does it follow that $f \rightarrow g$ as $x \rightarrow c'$?

Answer 1

Yes. Semi-formally, suppose we want $f(x)$ and $g(x)$ to be within $\varepsilon$ of each other in a punctured neighborhood of $x$ around $c'$. Then we find $\delta$ such that both $f(x)$ and $g(x)$ are within $\varepsilon/2$ of $c$ in a punctured neighborhood of $x$ around $c'$. Then, because $|f(x)-g(x)| \leq \varepsilon/2 + \varepsilon/2 = \varepsilon$, we are done.