Formally Evaluating a Limit

Question

The epsilon-delta definition of the limit says that $\lim_{x \to a}f(x)=L$ if for every number $\epsilon >0$ there is some number $\delta>0$ such that $|f(x)-L|<\epsilon$ whenever $0<|x-a|<\delta$.

Using this definition I've seen many examples asking to prove a limit is true. But, how would you use this to evaluate a limit?

Question: Ascertain $\lim_{x \to 3} x^2$ using the epsilon-delta definition of the limit (not prove that the limit is 9, but arrive at the limit is 9 without postulating).

Answer 1

Most likely, you'd prove some general rules about limits -- like the limit of a product is the product of the limits, etc. -- and then apply them to your question. Or, you could take a guess, and then use the definition to check if your guess is right. But, as you noticed, the definition isn't constructive -- you can't use it to construct the limit directly out of thin air. Nonconstructive definitions aren't uncommon in mathematics.

Answer 2

Note that $\epsilon-\delta$ definition is not used to calculate the limits but only to show some basic case and theorems which are used to calculate limits.

In this case by to use the definition we should assume a priori that

$$\lim_{x \to 3} x^2=9$$

and then show that $\forall\epsilon >0$ $\exists\delta>0$ such that $|x^3-9|<\epsilon$ whenever $0<|x-3|<\delta$.

The proof can be done by handling the inequality and finding $\delta$ which satisfies it.

Answer 3

In order to use $\varepsilon$-$\delta$, you first have to make a conjecture on what the limit should be. In this case the obvious conjecture is that the limit is $9$.

Then you must show that, for every $\varepsilon>0$, the inequality $|x^2-9|<\varepsilon$ is satisfied for $x$ such that $0<|x-3|<\delta$, for a suitable $\delta>0$.

Since $|x^2-9|=|x+3|\,|x-3|$ and you can assume $0<\delta<1$, for $0<|x-3|<\delta$ you also have $-1<x-3<1$, so $5<x+3<7$. Take $\delta=\min(1/2,\varepsilon/7)$ and you're done.

Actually, this is not necessary, because you can prove the function $x\mapsto x^2$ is continuous using general theorems (the identity function is continuous, the product of continuous functions is continuous,…). Thus this limit is $9$ by continuity.

In other cases you can apply different techniques such as “squeezing”, l'Hôpital, Taylor expansion and others.

Answer 4

You can't use that definition of the limit, to calculate it, since you need to know (or guess) the limit in order to use it. You can though use the Heine definition of the limit (sequential definition of the limit) which is equivalence to the epsilon-delta definition.

Take any sequence $x_n\to3$, then as $x\mapsto x^2$ is a continuous function (as a product of continuous function), $x_n^2\to3^2$.

Answer 5

Consider $|x^2-9| = |x-3||x+3|$. You estimate $|x+3| = |x-3+6| \le |x-3|+ |6| = |x-3|+6 < 1+6 = 7$ whenever $|x-3| < 1$. Thus if $|x-3| < 1 \implies |x^2-9| = |x-3||x+3| < 7|x-3|$. Also if $|x-3| < \delta\implies |x^2-9| < 7\delta$. Since you want $|x^2-9| < \epsilon$, you should take $7\delta = \epsilon$ which means $\delta = \dfrac{\epsilon}{7}$. Combining with $1$ above, you might take $\delta = \text{min}\left(1,\dfrac{\epsilon}{7}\right)$. With this choice of $\delta$, all inequalities above are true, and the statement that: if $0 < |x-3| < \delta$ then $|x^2-9| < \epsilon$ is true and this means the limit is $9$.

Answer 6

The definition of a limit is given in the form of a test so that it can only be used to check whether some number is the limit or not. It can not be used directly to evaluate limits. The definition of limit is used to prove certain theorems (algebra of limits, Squeeze theorem, standard limits etc) which can then be used to evaluate limits. Using these theorems to evaluate limits is a standard/common, formal and rigorous method.

In the current scenario you have to guess the limit somehow. First approach is plugging (because there is a happy and surprising fact that for most commonly occurring functions in calculus, the limit can be evaluated by plugging). Another approach is using calculator to find values of function when the independent variable takes values near point under consideration and find a pattern in the resulting values. Both these approaches lead to the guess that $9$ should be desired limit and then you can apply the definition to check if this guess is true. You should be able to do this on your own as it is not very difficult and probably your textbook has a similar solved example.