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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 …

Limit

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 $\epsilon_1 > 0$. $|g(x)- B|<\epsilon_2$ whenever $ 0 < |x-a| < \delta$, with $\epsilon_2 > 0$. 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: Using algebra, $ [f(x) – A] [g(x) – B] = f(x)g(x) – …

Limit

Limit of a Difference 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]. 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, $=\lim_{x\rightarrow a}f(x) + (-1)\lim_{x\rightarrow a}g(x)$ …

Limit

Limit of a Function Multiplied by a Scalar

In this post, I show that $ \lim_{x \rightarrow a} [c f(x)] = c \lim_{x \rightarrow a} [f(x)]$ if $c$ is a variable for any real number. To do this, I approximately follow the steps in reference [1]. Essential Background Information: Definition of a Limit: A function $f(x)$ approaches a limit $A$ as $x$ approaches $a$ if, and only if, for each positive number $\epsilon$ there is another, $\delta$, such that whenever $0 < |x-a| < \delta $ we have $|f(x) – A| < \epsilon$. That is, when $x$ is near $a$ (within a distance $\delta$ from it), $f(x)$ is near $A$ …

Limit

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 if, for each positive number $\epsilon$ there is another, $\delta$, such that whenever $0 < |x-a| < \delta $ we have $|f(x) – A| < \epsilon$. That is, when $x$ is near $a$ (within a distance $\delta$ from it), $f(x)$ is near $A$ (within a …