Formal Definition Of A Limit

The simplified definition of a limit states that f(x) approaches L as x approaches a. But what exactly does approach mean? How close does x get to a?

Definition §

The formal definition of a limit is:

For any arbitrary positive real number $ϵ$, there is a positive real number $\delta$ such that:

$$|f(x)-L|<ϵ⟺0<|x-a|<\delta$$

It’s a bit hard to grasp initially, but the key point is the second bound, $0<|x-a|<\delta$. Since $|x-a|$ is an absolute value, the only time it is equal to zero is when $x=a$. Therefore, for this inequality to hold, $x\ne a$ (which is what is defined casually anyway). Now, all we need to do is to transform one inequality to the other, as seen below:

1: Use in Proof

Using the formal definition of a limit, prove that ${\lim }_{x\to 1}2x=2$:

To prove this, we need to show that $|f(x)-L|<ϵ⟺0<|x-a|<\delta$ for $2x$ $|f(x)-L|=|2x-2|=2|x-1|<ϵ$ $2|x-1|=2\times |x-a|<ϵ$ $|x-a|<\frac{ϵ}{2}$ If we choose $\delta =\frac{ϵ}{2}$ then $|2x-2|<ϵ⟺0<|x-1|<\delta$ I.e. $|f(x)-L|<ϵ⟺0<|x-a|<\delta$. Therefore, ${\lim }_{x\to 1}2x=2$