﻿ LIMITS

LIMITS return

Limits are a very important concept in calculus.

First idea to understand is that the limit is the y or f(x) value not the x value.

Next we need to be concerned with the terminology.

example:   limx → 3 f(x) = 2x + 1

The above means that as the value of x gets closer and closer to x = 3 the value of f(x) approaches 7. Therefore the limit is 7. So if you substitue 2.9, 2.09, or 2.009 for x in the function the values of f(x)= 6.8, 6.98, 6.998. Getting closer and closer to 7. For the limit to exist the f(x) value must approach 7 when values greater than 3 such as 3.1, 3.01,.3.001 are substituted in for x. If you substitue these values of x into f(x) you find the value of f(x) are 7.2, 7.02, 7.002 which is getting closer and closer to 7 as x get closer to 3. Therefore the limit exist and it is 7.

f(x) = 2x + 1

 x = 2.9 2.99 2.999 3 3.001 3.01 3.1 y = 6.8 6.98 6.998 7 7.002 7.02 7.2

We are only concerned with what happens to the function as the value of x approaches 3.

Some additional notation - In the above example when the values of x approaches 3 but are less than 3 we say that x is approaching 3 from the left and write it as  limx → 3-  f(x) = 2x + 1   and when the value of x is greater than 3 we say that x is approaching x from the right and write it as   limx → 3+   f(x) = 2x + 1