Limits

A limit is the behavior of a function as it approaches a certain point. This can be from the left or right.

In order to evaluate the limit, you should first plug in the number that the function is approaching. If you don't get a value and instead get undefined, you should use different methods depending on the type of function to solve for the limit. You should also try numbers on both sides of the number the function is approaching. This will verify if the limit is actually a true limit.

There are several ways to evaluate:
-Factor and cancel
-Rationalize using the conjugate (if there are square roots)
-Combine fractions

Factoring and Canceling

 lim     2x²-13x+15
x→5      x²+x-30

           2x²-10x-3x+15
             x²+6x-5x-30

           2x(x-5)-3(x-5)
           x(x+6)-5(x+6)

           (2x-3)(x-5)
            (x-5)(x+6)

           (2x-3)(x-5)
            (x-5)(x+6)      

            (2x-3)
             (x+6)

           (2(5)-3)
            (5+6)

           10-3
            11

lim     =    7
x→5        11

Rationalize using the Conjugate

lim      √(5x-6) -2
x→2      x²+x-6

           √(5x-6) -2  •√(5x-6) +2
             x²+x-6     •√(5x-6) +2

                        5x-10            
           (x²+x-6)(√(5x-6) +2)

                        5(x-2)             
           (x-2)(x+3)(√(5x-6) +2)

                        5             
          (x+3)(√(5x-6) +2)

                        5             
          (2+3)(√(5(2)-6) +2)

 lim      =  1
x→2        4

Combining Fractions
lim        1   _       1    
x→0     x²         x⁴+x²

             1( x²+1)    _       1    
             x²( x²+1)          x⁴+x²

                  x²+1    _       1    
              x⁴+x²          x⁴+x²

               x²    
            x⁴+x²         

                 x²       
           x²( x²+1) 

                  1   
            x²+1

lim      =
1
x→0


Piecewise Functions




In this graph of a piecewise function, you can tell where roots will and won't exist. Roots only exist when the graph approaches the same point from the left and the right. If the graph approaches different points when coming from different directions, the limit doesn't exist.

 For this graph, the limit as the function approaches -1 from the left is -1. The limit as the function approaches -1 from the right is also -1. This means -1 is a limit of the graph.

For this graph, the limit as the function approaches 2 from the left is 4. The limit as the function approaches 2 from the right is 3. Due to the fact that the limits are different, the limit does not exist.

Infinite Limits
Infinite limits are where the limit of a function is either ∞ or -∞.
An example of this type of function is:

lim      2
x→0   x²

You can enter any number close to 0 into the function and you will keep getting a large number. You can do this by approaching from the right or left. This means the limit as the function approaches 0 is ∞.