Math 1241, Spring 2014 Section 3.1, Part Two

1. Math 1241, Spring 2014Section 3.1, Part Two Infinite Limits, Limits “at Infinity” Algebraic Rules for Limits

2. Infinite Limits • If x is close to zero, then the function is close to what number? Here is the graph:

3. Infinite Limits • IMPORTANT: DOES NOT EXIST!!! • There is a reason for this. As x approaches 0, the function value keeps getting larger, and never approaches any particular value. • Notation: • But you CANNOT treat the infinity symbol as though it were an ordinary number.

4. Examples of Infinite Limits Convince yourself (possibly by drawing a graph) that the following are true: For the left-sided limit, the means that the function value continues to decrease, and does not approach any particular value.

6. Limits “at Infinity” • On the previous graph, what happens to the value of f(x) as x gets “larger and larger?” • On the graph: Further and further to the right. • Similar question: what happens to the value of f(x) as x gets “more and more negative?” • On the graph: Further and further to the left. • In previous courses, these questions were related to horizontal asymptotes of the graph.

7. Limits “at Infinity” The notation: means, “As the value of x gets larger and larger, the value of f(x) approaches the number L.” In similar fashion: means, “As the value of x gets more and more negative, the value of f(x) approaches the number L.”

8. Example:

9. Infinite Limits “at Infinity” • We can also have infinite limits “at infinity.” For example: • This means, “As the value of x gets larger and larger, the value of becomes more and more negative.” See the graph on the next slide. • NOTE: If a limit “equals” + or , that limit DOES NOT EXIST. The notation allows us to indicate why the limit does not exist.

11. Algebraic Rules for Limits • For most “ordinary” algebraic functions, you can “plug in x = a” to evaluate . • In particular, this works for: • Polynomials (example: ) • Rational functions, except when the denominator is zero: (example: for ) • Exponential functions (example: ) • Logarithms (example: )

12. Simple Algebraic Examples Evaluate . • Solution: Since is a polynomial, you can evaluate the limit by plugging in x = 3. • . • We can confirm this with a graph, see the next slide.

14. Exercises Evaluate the following limits, and compare your results with the previous graph. These should be very easy exercises!

15. Exercise: Rational Functions Let Evaluate the limits: The first limit should be very easy. The second requires more work.

16. Solutions • Since is a rational function (ratio of two polynomials), we can evaluate by plugging in x = 1: • What happens when you plug in x = 2 ?

18. Many graphing programs do not detect the “hole in the graph” when x = 2. • When our function has a zero denominator, we can try to factor numerator/denominator, and hope that the zero factor cancels. • HINT: In this case, the numerator and denominator are zero at x = 2, so there should be a factor of (x-2).

19. When evaluating the limit as , we may assume . This allows us to cancel the zero factor from the denominator. • If you graph , you will see that it has the same graph as the function above (with the hole filled in at ). • We can now plug in x = 2, giving a limit of .

20. An Important Result • For this type of scenario, we have the following: If whenever , then • In other words, the value of (even if it is undefined) does not have any effect on the value of . • Note: This is Rule #7, pg. 128. We saw a graphical version of this last time.

21. For more complicated functions, we can often evaluate limits with the following rules (pg. 128)

22. Roots/Fractional Exponents • We can usually “take the limit symbol through the radical sign,” but we must be careful with even roots (including square roots). • If n is an odd positive integer, then provided that exists. • If n is even, the above rule works when .

23. Example: The idea is to use the limit rules to break this down into limits of more simple functions. (provided that both limits on the right exist) • (why?) (provided the limit under the radical exists and is positive) • (why?)