# The Limit Definition

Having discussed how you can compute limits, I want to examine the definition of a limit in more detail.

Example. Why is it necessary to be careful? Suppose you're trying to compute . You might think of drawing a graph; many graphing calculators, for instance, produce a graph like the one below:

(I produced the graph using a program called Mathematica.)

It looks as though the graph is dropping down to 0 near . You might guess that the limit is 0. In fact,

It's possible to justify this algebraically once you know a little about limits of trig functions.

Pictures can be helpful; so can experimenting with numbers. In many cases, pictures and numerical experiments are inconclusive or even misleading; at best, they suggest rather than {\it prove}. Sometimes understanding requires more precision.

Informally,

means that can be made arbitrarily close to L for all x's sufficiently close to c.

This statement is like a guarantee. Think of making parts in a factory. Your customers won't buy your parts unless they meet certain specifications. So you might guarantee that your parts will be within 0.01 of the customer's specification.

Likewise, to say that you must be able to guarantee that you can make fall within (say) 0.01 of L. But you have to do more: You have to be able to make fall within any positive tolerance of L --- 0.0001, 0.0000004, and so on, no matter how small.

Another way to think of this is as meeting a challenge:

Challenge: "I challenge you to make fall within 0.0005 of L."

Your response: "I guarantee that every x within 0.003 of c (except perhaps c itself) will give an that is within 0.0005 of L."

To prove that , you must be able to meet the challenge no matter what positive number is used in place of 0.0005.

By the way, notice that is excluded in my guarantee. Reason: As I noted earlier, in computing , you only consider what happens as x approaches c, not what is.

Example. You can see by plugging in that

How close should x be to 4 to guarantee that is within 0.01 of 7?

Remember that

I want to be within 0.01 of 7. This means

So

The last inequality says that the distance from x to 4 should be less than . So if x lies within of 4, I can guarantee that will be within 0.01 of 7.

Can you see that if I'm challenged to make lie within 0.00001 of 7, I should make x lie within of 4? Just replace 0.01 with 0.00001 in the discussion above.

And similarly, I can make lie within any tolerance FOO of 7 by making x lie within of 4.

This shows that I can meet any challenge, since I can just take the challenge tolerance and plug it in for FOO. This proves that

Example. The graph of a function is shown below.

I claim that . Suppose I'm challenged to make fall within 0.5 of 3. That is, I want my y-values to fall within the grey strip in the picture.

On the right side of 4, the graph stays within the grey strip as far as 4.25; on the left side of 4, the graph stays within the grey strip as far as 3.

I'll use the closer of the two values, which is 4.25. Now 4.25 is 0.25 units from 4, so my answer is: If x is within 0.25 of 4, then will be within 0.5 of 3.

If I can meet such a challenge with any positive number in place of 0.5, then I will have proved that .

Example. (Disproving a limit) Consider the function whose graph is show below.

Suppose that Calvin Butterball thinks that . I can use the limit definition to disprove it; to do so, I'll make a challenge that Calvin can't meet.

I challenge Calvin to make fall within 0.5 of 4. This means that he must find a range of x's around 3 so that the corresponding part of the graph lies within the grey strip shown below:

You can see that there's no way to do this. (Note: He's not allowed to use alone. Remember that what the function does at has no bearing on the value of the limit.)

Since this challenge can't be met, . In fact, is undefined.

Example. Suppose

It is true that . How close should x be to 1 in order to guarantee that will be within 0.0008 of 3?

From the left side, I'd need

This is the same as , or . The last inequality says that x should be within 0.0004 of 1.

From the right side, I'd need

This gives , which means that x should be within 0.0002 of 1.

To satisfy the two requirements at the same time, I'll use the smaller of the two numbers. Thus, if x is within 0.0002 of 1, then will be within 0.0008 of 3.

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