Calculus/Formal limits

Formal definition of limit (optional)

In preliminary calculus, the definition of a limit is probably the most difficult concept to grasp. If nothing else, it took some of the most brilliant mathematicians 150 years to arrive at it.

The intuitive definition of a limit is adequate in most cases, as the limit of a function is the function of the limit. But what is our meaning of "close"? How close is close? We consider the following limit of a function:

$\underset{x\to 0}{\lim }f(x)=\underset{x\to 0}{\lim }\frac{\sin (x)}{x}.$

You will have noted that f(0) = 0/0, and this is undefined. But the limit does exist and it equals 1. But how do we convert that intuition into mathematical sense?

We call L the limit of f(x) as x approaches c if for every positive number ε, there exists a number δ such that

$|f(x)-L|<ϵ$

whenever

$0<|x-c|<\delta .$

Understanding the parallels between the two definitions above is important: instead of saying ' f(x) approximately equals L ', the formal definition says that 'the difference between f(x) and L is less than any number epsilon'.

$\underset{x\to c}{\lim }f(x)=L$ if and only if $\underset{x\to c^{-}}{\lim }f(x)=\underset{x\to c^{+}}{\lim }f(x)=L$

$\underset{x\to c^{-}}{\lim }f(x)$ means as x approaches c from the left, and $\underset{x\to c^{+}}{\lim }f(x)$ as x approaches c from the right.

See how to make formal proofs.

Notice that we have not yet defined ' x approaches c ' yet; we will discuss this later.

Limits are often written

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

which may be read "the limit of f(x) as x approaches c".

Examples

1) What is the limit of f(x) = x + 7 as x approaches 4?

There are two steps to answering such a question; first we must determine the answer -- this is where intuition and guessing is useful, as well as the informal definition of a limit. Then, we must prove that the answer is right. For this problem, the answer happens to be 11. Now, we must prove it using the definition of a limit:

Informal: 11 is the limit because when x is roughly equal to 4, f(x) = x + 7 approximately equals 4 + 7, which equals 11.

Formal: We need to prove that no matter what value of ε is given to us, we can find a value of δ such that

$|f(x)-11|<ϵ$

whenever

$|x-4|<\delta .$

For this particular problem, letting δ equal ε works (see choosing delta for help in determining the value of delta to use). Now, we have to prove

$|f(x)-11|<ϵ$

given that

$|x-4|<\delta =ϵ.$

Since |x - 4| < ε, we know
|f(x) - 11|  = |x + 7 - 11| = |x - 4| < ε, which is what we wished to prove.

2) What is the limit of f(x) = x² as x approaches 4?

Formal: Again, we pull two things out of thin air; the limit is 16 (use the informal definition to find the limit of f(x)), and δ equals √(ε+16) - 4. Note that δ is always positive for positive ε. Now, we have to prove
$|x^2-16|<ϵ$
given that
$|x-4|<\delta =\sqrt{ϵ+16}-4$.

We know that |x + 4| = |(x - 4) + 8| ≤ |x - 4| + 8  < δ + 8 (because of the triangle inequality), thus

$\begin{array}{ccc}|x^2-16|& =& |x-4|\cdot |x+4|\\ \\ & <& (\delta )\cdot (\delta +8)\\ \\ & =& (\sqrt{16+ϵ}-4)\cdot (\sqrt{16+ϵ}+4)\\ \\ & =& (\sqrt{16+ϵ}{)}^2-4^2\\ \\ & =& ϵ\end{array}.$

3) Show that the limit of sin(1/x) as x approaches 0 does not exist.

Suppose the limit exists and is l. We will proceed by contradiction. Assume that $l\ne 1$, the case for $l=1$ is similar. Choose $ϵ=l-1$, then for every $\delta >0$, there exists a large enough n such that $0<x_0=\frac{1}{\pi /2+2\pi n}<\delta$, but $|sin(1/x_0)-l|=|1-l|=ϵ$ a contradiction.

The function sin(1/x) is known as the topologist's curve.

4) What is the limit of $xsin(1/x)$ as x approaches 0?

It is 0. For every $ϵ>0$, choose $\delta =ϵ$ so that for all x, if $0<|x|<\delta$, then $|xsinx-0|<=|x|<ϵ$ as required.

5) Prove that the limit of 1/x as x approaches 0 does not exist?

6) Prove that the limit of f(x)={x, when x is rational; 0, when x is irrational} as x approaches c for ALL (c not equal to zero, c is real number) does not exist?