1. 3. Limits and infinity

2. Vertical Asymptotes (VA)
If then x=a is a VA of f(x)
To find VA algebraically – set denominator = 0

3. Example 1 – Find VA

4. Finding limits on either side of a VA
Plug in a value greater than VA and see if you get a positive or negative number
If positive then
If negative then
Now plug in a value less than VA and see what sign you get
If it is a one-sided limit, you only need to do one of these

5. Example 2

6. Horizontal asymptotes (HA)
If or , the line y = b is an HA
A horizontal asymptote is NOT a discontinuity
A graph can cross its HA any number of times (although they don’t have to)
An HA only describes what may happen to a function at its extreme endpoints (its end behavior)

7. Finding HA algebraically
Find the highest exponent in the numerator and denominator
If they are the same, divide coefficients to get HA
If the higher one is on bottom, y=0 is HA
If the higher one is on top, there is a slant asymptote (do long division to find it)
When you take the limit as x approaches infinity or negative infinity for this third type, the answer will be either infinity or negative infinity. To find out which one, plug in a positive or negative value and see what type of answer you get.

8. Example 3 – find horizontal asymptote

9. Example 4

10. Non-explicit degrees
Sometimes the degrees of the numerator and denominator are not explicit (like when they are under a radical)
Example 4
What about

11. Growth rates
Different functions grow at different rates for large values
Log functions grow slowly
Polynomials grow faster by order of degree
Exponential functions grow faster than any polynomial
Factorials are next
xx are the king of growth
You can find limits at infinity by analyzing respective growth limits

12. Example 6

13. Limits that don’t involve infinity but are nonetheless special
Knowing the transformations of this and knowing where the jump occurs (the value that yields 0/0) will allow you to answer limit questions about this function

14. Example 6
Evaluate the following

15. Limits that don’t involve infinity but are nonetheless special
Piecewise functions
Example 8

16. Summary
If there is a limit as x approaches , there is a horizontal asymptote at the limit
If the limit equals , there is a vertical asymptote at the x value