# 241.19 the e d definition of a limit.

• MHB
• karush
In summary, in order to prove the statement using the $\epsilon,\delta$ definition of a limit, we need to find a value of $\delta$ that satisfies the condition $0<|x-1|<\delta$ and ensures that $\left|\dfrac{2+4x}{3}-2\right| <\epsilon$. This can be achieved by setting $\delta = \dfrac34\epsilon$.
karush
Gold Member
MHB
prove the statement using the $\epsilon,\delta$ definition of a limit.
$$\lim_{{x}\to{1}}\frac{2+4x}{3}=2$$
so then
$$x_0=1\quad f(x)=\frac{2+4x}{3}\quad L=2$$
now
$$0<|x-1|<\delta\quad\text {and}\quad\left|\frac{2+4x}{3}-2\right| <\epsilon$$
then
$$\left|\frac{2+4x}{3}-\frac{6}{3}\right|=\left|\frac{4x-4}{3}\right|$$
$$=\frac{4}{3}|x-1|=|x-1|<\frac{3}{4}\epsilon$$
finally
$$\left|\frac{2+4x}{3}-2\right| =\frac{4}{3}|x-1|<\frac{4}{3}\delta =\frac{4}{3}\left(\frac{3}{4}\epsilon\right) =\epsilon.$$

Last edited:
karush said:
prove the statement using the $\epsilon,\delta$ definition of a limit.
$$\lim_{{x}\to{1}}\frac{2+4x}{3}=2$$
so then
$$x_0=1\quad f(x)=\frac{2+4x}{3}\quad L=2$$
now
$$0<|x-1|<\delta\quad\text {and}\quad\left|\frac{2+4x}{3}-2\right| <\epsilon$$
Given $\epsilon>0$, we need to find $\delta>0$ such that $\left|\dfrac{2+4x}{3}-2\right| <\epsilon$ whenever $0<|x-1|<\delta$.​
karush said:
then
$$\left|\frac{2+4x}{3}-\frac{6}{3}\right|=\left|\frac{4x-4}{3}\right|$$
$$=\frac{4}{3}|x-1|$$
That calculation should then continue $$\left|\frac{2+4x}{3}-\frac{6}{3}\right|=\left|\frac{4x-4}{3}\right|$$
$$=\frac{4}{3}|x-1| < \frac43\delta.$$ In order for that to be less than $\epsilon$, we can take $\delta = \dfrac34\epsilon.$

So your solution is essentially correct, but the wording needs to be clarified.

## What is the definition of a limit?

The definition of a limit, as given by 241.19 the e d, is the value that a function approaches as the input value approaches a specific point.

## How is the limit of a function calculated?

The limit of a function can be calculated by plugging in values closer and closer to the specific point and seeing what value the function approaches.

## What is the purpose of finding the limit of a function?

The limit of a function helps us understand the behavior of the function at a specific point and can be used to determine continuity and differentiability.

## Can a function have multiple limits?

Yes, a function can have multiple limits depending on the direction from which the input value approaches the specific point. These are known as the left-hand and right-hand limits.

## What is the difference between a limit and a value of a function?

The limit of a function represents the behavior of the function at a specific point, while the value of a function represents the output of the function at that specific point.

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