Epsilon Delta Definition of a Limit

Math Calculus

Definition

When

$$\underset{x\to c}{\lim }f(x)=L$$

For $ϵ>0$ and $\delta >0$, there is a value $\delta$ for every value of $ϵ$ such that

$$\begin{gathered} 0<|x-c|<\delta \\ 0<|f(x)-L|<ϵ \end{gathered}$$

Proving a Limit

Let’s prove:

$$\underset{h\to 0}{\lim }\frac{(x+h{)}^2-x^2}{h}=2x$$

Let:

$$\begin{gathered} 0<|\frac{(x+h{)}^2-x^2}{h}-2x|<ϵ \\ 0<|\frac{(x+h{)}^2-x^2}{h}-2x|<ϵ \\ 0<|\frac{(x^2+2xh+h^2-x^2)}{h}-2x|<ϵ \\ 0<|\frac{2xh+h^2}{h}-2x|<ϵ \end{gathered}$$

Remember $ϵ>0$:

$$\begin{gathered} 0<|2x+h-2x|<ϵ \\ 0<|h|<ϵ \end{gathered}$$

We have to prove for every $ϵ$:

$$\begin{gathered} 0<|h-0|<\delta \\ 0<|h|<\delta \end{gathered}$$

These two inequalities are the same, so they are easily satisfied just by setting:

$$\delta =ϵ$$

Graphical Explanation

https://www.desmos.com/calculator/tucchymbrq