How To Find The Limit

What is a limit? Limit means to find what value y (or f(x) ) is as x approaches a certain number but can never quite reach the number. One way to find the limit is by the SUBSTITUTION METHOD.

For example, the limit of the following graph is 0 as x approaches infinity, because the graph approaches 0:

Sample A: Find the limit of f(x) = 4x as x approaches 3.

Steps:

1) Replace x for 3.
2) Simplify.

f(x) = 4x becomes f(3) = 4(3) = 12.

So, the limit of f(x) = 4x as x approaches 3 is 12.

Sample B: Find the limit:

lim	x^2 + 5x - 3 =
x-->1

Follow the same steps above.

lim (1)^2 + 5(1) - 3 = 1 + 5 - 3 = 6 - 3 = 3.

So, the limit of x^2 + 5x - 3 as x approaches 1 is 3.

HOWEVER, the substitution method will not always work.
For Sample C below, you must factor the numerator first BEFORE applying the substitution method.

Sample C:

lim	6x^2 - 7x
x-->0	x

NOTE: If we substitute 0 for x in Sample C, we will create division by zero which DOES NOT EXIST or is UNDEFINED.
This is the reason factoring MUST be our first step in this sample.

Factoring 8x^2 - 7x becomes x (8x - 7)/x.

We can now cancel x in the numerator and denominator. When we cancel BOTH variables x, we are left with 8x - 7. Now, we can substitute 0 for x in 8x - 7 to find the limit.

So, 8(0) - 7 = 0 - 7 = -7.

The limit of 8x^2 - 7x as x approaches 0 is -7.

Note: Even though we were able to simplify the function in Sample C by factoring, we need to represent the original function in the graph. We must leave a hole in the graph at point (0, -7) because the original function DOES NOT EXIST when x = 0.
In other words, if x = 0 in the original function, we will end up with division by zero, which is undefined or simply put DOES NOT EXIST. Only after factoring, in some cases, can we then apply substitution to find the limit.

By Mr. Feliz
(c) 2004