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: lim_{x → 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 lim_{x → 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 lim_{x → 3}+
f(x) = 2x + 1