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