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 $, $ …

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Limit of the Product of Two Functions

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]. Known:  Using the definition of a limit, $|f(x) – A|<\epsilon_1$ whenever $ 0 < |x-a| < \delta$, with …

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Limit of a Sum of Two Functions

In this post, I show that $ \lim_{x \rightarrow a} [f(x) + g(x)]$ is equal to $\lim_{x \rightarrow a}f(x) + \lim_{x \rightarrow a} g(x)$ 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: Using the definition of a limit, the objective is …

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Limit of a Constant

In this post, I prove that $ \displaystyle \lim_{x \rightarrow a} c = c $ if $c$ is a variable for any real number. To do this, I approximately follow the outline from reference [1]. Essential Background Information: Definition of a Limit: A function $f(x)$ approaches a limit $A$ as $x$ approaches $a$ if, and only …

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