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# Write a proof for each limit using the $\varepsilon-\delta$ definition of a limit. $\lim _{x \rightarrow 3}(4 x)=12$

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$\text{\color{#4257b2}We would like to proof for the following limit by using the\ (\delta,\ \epsilon) definition of limit.}$

$\color{Brown}\lim\limits_{x\to3}\ (4x)=12$

$\lozenge$\ \ $\text{\underline{\bf{Solution:}}}$

Assume that \ \ $\delta>0,\ \ \ \ \ \ \epsilon>0$

There is a general rule in the definition of limit:

$0<|x-a|<\delta\ \ \ \Rightarrow\ \ \ |f(x)-L|<\epsilon$

Apply this rule for the given data as follows:

$|x-3|<\delta\ \ \ \Rightarrow\ \ \ |4x-12|<\epsilon$

Use GCF property as follows:

$|x-3|<\delta\ \ \ \Rightarrow\ \ \ |4(x-3)|<\epsilon$

$|x-3|<\delta\ \ \ \Rightarrow\ \ \ 4|x-3|<\epsilon$

Divide the right side by $(4)$ as follows:

$|x-3|<\delta\ \ \ \Rightarrow\ \ \ |x-3|<\dfrac{\epsilon}{4}$

Since the absolute value in the both side are equal, so the result values are equals too:

$\because\ \ |x-3|=|x-3|\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \therefore\ \ \ \delta=\boxed{\ \dfrac{\epsilon}{4}\ }$

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