Calculation of Limits

Mathematical concept of limits will not be the same as yours, which may be casual. If we want to find the limit of f(x) when x tends to a, we don't mean to find f(a). We should always bear this in mind when calculating limits.

Consider a function f(x):

Find the limit of f(x) when x tends to 2.

I want to emphasize that we are NOT finding f(2).

Let's start from x=1 and 3. And we construct a table with values of x and f(x)

x 1,3 1.5,2.5 1.9,2.1 1.999,2.001
f(x) 2,0 1.5,0.5 1.1,0.9 1.001,0.999

From this table, we can see that f(x) TENDS to 1 (not -7), when x TENDS to 2.

The method of finding the limits is that:

(This is not the epsilon-delta definition but the same in meaning/ concept)

In this case, we choose the small variation of x as 1, then 0.5, 0.1, 0.001. Since all f(x) in this case equals to 3-x, then the limit should be 1.

This also introduces an example of not continuous function: limit of f(x) when x tends to a is not equal to f(a). Another example is 1/x.

Another example:

In this example, we used some algebraic operations to attain the value. The following can be left as exercise, since before the next step, the foundation is very important.