1. 2.2 Limits at Infinity: Horizontal Asymptotes
Infinite Limits: Vertical Asymptotes

2. Limits when x Increases Without Bound
Let f be a function defined on some interval (a, ∞). Then
Means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large.

3. Formal Definition

4. Limits when x Decreases Without Bound
Let f be a function defined on some interval (-∞,a). Then
Means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large negative.

5. Formal Definition

6. Horizontal Asymptote
The line y = b is called a horizontal asymptote of the curve y = f(x) if either or

7. Example
Solution

9. Crossing Horizontal Asymptotes
Unlike vertical asymptotes that can never be crossed, horizontal asymptotes are about behavior for LARGE x values.
Behavior close to the origin may cross the horizontal asymptote.
What is important is what happens as x gets very large in the positive or negative direction.

10. Multiple Horizontal Asymptotes
If the limits are different depending on the direction, then there may be more than one asymptote –
One asymptote will occur as x∞
Another may occur as x - ∞
An example would be:

11. Limit Laws
The limit laws used for definite numbers can also be used for finding limits as x±∞.

12. Infinite Limit of 1/x
It is important to note that
and

13. Infinite Limit of when r > 0
If r > 0 is a rational number, then
If r > 0 is a rational number such that
is defined for all x, then

14. Agenda – Monday, August 13th
Finding limits analytically hw quiz
Finish notes on limits at infinity & horizontal asymptotes
2.1: 8,10,12,14,16,18,20,22,24,26,28,31,32, 33,34,35,36,37,38,39,40,41,42,43,44,49,50,51,54,58,63 due 08/14

15. Finding Horizontal Asymptotes of Rational Functions
Divide both numerator and denominator by the variable with the highest exponent in denominator.
NOTE: If you must deal with a radical, remember that when you divide inside the radical you are dividing by the highest exponent times the index
Simplify.
Take the limit of each term in the numerator and denominator.

16. Example
Solution
Find the highest degree n of the terms, then multiply the numerator and the denominator by

17. Simplify
Take limit of each
term

18. Example
Solution

19. Example
Solution

20. Example

21. Beware of Indeterminate Forms
Indeterminate forms involve ∞’s and 0’s from which no conclusion can be drawn. Examples are:
∞-∞
∞/∞
0●∞
∞^0
1^∞
0/0
0^0

22. Going Beyond the Indeterminate Form
When encountering an indeterminate form, it is necessary to try “trickery” to find the limit.
Try “tricks” learned earlier when finding limits such as rationalization, reduction, squeeze theorem, etc.
Ask: Is either term increasing more quickly? Then that term will dominate the answer.

23. Using Asymptotes and Intercepts to Sketch Graphs
Find the vertical asymptote and draw that with a dotted line. Label.
Find the Horizontal asymptote and draw that with a dotted line. Label.
Find the x-intercepts using algebra (factoring, quadratic formula etc.). Plot these.
Remember your rules of multiplicity:
If the intercept factor is raised to a even power it touches x and returns,
If it is raised to an odd power, it passes through.

24. Vertical Asymptotes: Infinite Limits
When the limit moves unchecked to either + or - ∞, the limit does not exist.
However, we will use + or - ∞ to denote the function is increasing (decreasing) without bound in either the + or - direction.

25. What happens as x approaches 0?
As x approaches 0 from the left, the equation values increase without bound
As x approaches 0 from the right, the equation values increase without bound.

26. Example

27. Vertical Asymptote
If the function increases without bound toward either + or - ∞ when x approaches a from either the right or left–hand side, the imaginary line at x = a is a vertical asymptote.

28. Does this function have a vertical asymptote?
Yes, at x =0

29. Does this function have a vertical asymptote?
Yes, at x =0

30. Example
Solution
So x=2 is definitely a vertical asymptote

31. Final Thought
It is important to remember that infinite limits are not true limits, they only give an indication of a function’s behavior.