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Limit

Limit of a Ratio of Two Functions

In this post, I show that $\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x \rightarrow a}g(x)}$ given that $\lim_{x\rightarrow a}f(x)=A$, $\lim_{x \rightarrow a} g(x) = B$, $B \ne 0$ and $g(x) \ne 0$. To do this, I approximately follow the steps in reference [1]. Known: From the the definition of a limit, Whenever $ 0 < |x-a| < \delta $, $ |f(x) – A| < \epsilon_1$ with $\epsilon_1 > 0$. Whenever $ 0 < |x-a| < \delta $, $ |g(x) – B| < \epsilon_2$ with $\epsilon_2 > 0$. Objective: The objective is to directly show that $ \lim_{x \rightarrow a} \frac{f(x)}{g(x)} =  \frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow …