# Question about epsilon-delta definition of limits.

In Chapter 1: Functions and limits, 1.7 The Precise Definition of a Limit,

Let $f$ be a function ... the limit of $f(x)$ as $x$ approaches $a$ is $L$, and we write $$\lim_{x\to a }f(x)=L$$ if for every number $\epsilon>0$ there is a number $\delta>0$ such that if $0<|x-a|<\delta$ then $|f(x)-L|<\epsilon$.

OK, let's assume this is the case and that this definition is perfectly true, but:

• Then how do you go about determining this $L$ by the definition? For example if I wanted to compute $\lim_{x\to0}\frac{\sin x}{x}$ how can I determine this $L$ from the definition (without pre-knowledge of the limit)? (this is related to the question below)
• Can we find a suitable $\delta_1$ such that $|f(x)-L_1|<\epsilon$ and a different $\delta_2$ such that $|f(x)-L_2|<\epsilon$? If so then what is right $L_1$ or $L_2$?

I think if those issues are cleared to me then the rest will also be clear.

• And of course he can't prove his definition is correct. It's a definition. Hand-waving is always used to justify definitions. – Thomas Andrews Sep 23 '14 at 19:36
• I'm more talking about why we choose this particular definition given those issues? Or how are those issues resolved to support the definition? – user178276 Sep 23 '14 at 19:38
• Read on, and you will see how to compute $L$ in special cases. As noted, this is the definition, and it will give you the results you expect when you expect them. For your second issue, when $\epsilon$ beomes arbitrarily small, there can only be at most one limit $L$. Indeed, supposing there are two means $|L_1-L_2|<\epsilon$ for all $\epsilon$, so $L_1=L_2$ – Shakespeare Sep 23 '14 at 19:39
• @user178276 I know this is old but I wanted to address your question "about why we choose this particular definition". As mathematicians, we can all define things in whatever way we want. I could make my own definitions for anything! The biggest thing, though, is finding meaning in these definitions. For example, I could define something to be the number of letters used in an equation, but the question is: "so what?" How does that help in the world of math? As it turns out, our definition of a limit helps build the foundation of calculus, which can be used to solve countless problems. – Sultan of Swing Dec 11 '14 at 12:32

The definition doesn't show you how to determine $L$. There is no general way to determine $L$. This definition only shows that at most one $L$ exists (see below.) If there was a general way to determine $L$, calculus would be a nothing-burger.
For any particular $\epsilon$ you can find two $L_1$ and $L_2$ and $\delta_1$ and $\delta_2$, but there is no way for it to work for all $\epsilon.$ The definition says that for all $\epsilon>0$. This is provable from the definition:
If $L_1\neq L_2$ let $\epsilon=|L_1-L_2|/2$. Try to pick $\delta_1$ and $\delta_2$ for this $\epsilon$. One can prove that this is not possible.
• @user178276: If there are two limits, $L_1$ and $L_2$, then let $0 < \epsilon < |L_1 - L_2|/2$. Pick your $\delta_1$ and $\delta_2$. Then if $|x - a| < \min(\delta_1,\delta_2)$, we have $f(x)$ is closer than $\epsilon$ to both $L_1$ and $L_2$, which is impossible. – Nick Matteo Sep 23 '14 at 19:54