1. The concept of limit

A plain definition: means f(x) can be made as close to L as desired by making x close enough to a. This is quite to the intuition. Let’s see an example.

We know that 2² = 2 x 2 = 4. As a result, if we just plug in 2 we get 4. Yes, 4 is the answer, but math is all about concept. It’s important to know why. According to the plain definition above, x can’t reach 2 if we are calculating the limit above. However, we can make it as close to 2 as we want.

Say if x = 1.99, then x² = 1.99 x 1.99 = 3.9601. We can make it even closer.

If x = 1.999 then x² = 3.996001.

If x = 1.9999 then x² = 3.99960001.

You see a pattern here. As we make x closer to 2, the value we get is increasingly close to something, which is 4. But no matter how we make x close to 2, the value will not equal to 4 or exceed 4. In this sense we say that

This is the right way of thinking limit. Plug in number is not.

Notice that limit may not exist. If a limit does not exist, this means that limit does not equal to anything in particular. For example . When x approaches 0, will change radically and will not close on a particular value.

2. Working with limits

There’re a few rules to help us calculate limit.

(1) the limit of a sum is the sum of the limits:

(2) the limit of a product is the product of the limits, provided the limits exist:

(3) the limit of a quotient is the quotient of the limit, provided the limits exist and the denominator is not zero:

Using these rules we can calculate most limits. See an example below.

We know that x²-1 = (x+1)(x-1). If x ≠ 1 then we can simplify the above to x+1.

(because x->1 makes x ≠ 1 true)

3. Infinity

means that f(x) is as large as you like provided x is close enough to a.

means that f(x) is as close as you want to L provided x is big enough.

Notice that ∞ is not a number. Lim = ∞ should be understood as a statement.

For example, we want to calculate the limit below:

We can’t use quotient rule directly because . ∞ is not a number which means the limit does not exist. We have to work our way round.

This is a common technique for calculating limit, but never mind. You won’t have much chance to work with limit directly.

4. Why do we need limit?

This is a question fair enough. Why do we need the concept of limit? There should be a bigger purpose, and there it is. Limit serves as the founding block for differentiation and integration. By founding block I mean two things. First, without the concept of limit differentiation and integration will not stand. Second, limit is such a low-level concept that we normally will not use limit directly in problems. Rather we’ll use high-level ideas and notations to make our life easier.

What’s important is that you understand the concept of limit.