2.4 Limits

Expressing Limits
The function F has the limit L as x approaches a, written

if the value f(x) can be made as close to the # L as we please by taking x sufficiently close to (but not equal to) a. This means that the limit is the number you are approaching as x gets closer to a.

Evaluating Limits
By Graphing

(The Limit as X approaches a of F(x) is L)

If the equation had a hole where L is the limit would still be the same.

.

(Infinite Limit is true by both sides, increasing without bounds.)

DNE = Does not exist

Your fingers do not meet when tracing.
BUT if you are looking for or it is possible.

Positive and negative signs indicate direction. Positive = traveling from right to left and Negative = traveling from left to right.

In this case the limits would be:

and

Evaluating Limits

Direct Substitution

To find the limit without looking at a graph. I plugged in 2 for the equation. 2 squared plus 6 equals 10. 10 is the limit as x approaches 2.

If you have a hole in your graph than you want to try and see the removable discontinuity.

transforms to

transforms to


now you can plug in 2 for the equation 4(x+2).
The limit is 16. The limit of 4(x+2) as x approaches 2 is 16.

Limits at Infinity

Rules:



1. If the degree of P(x) is greater than the degree of Q(x) , then the limit will be + or - infinity.
2. If the degree of P(x) is less than the degree of Q(x), then the limit will be a horizontal asympotote at 0.
3. If the degree's of the function are a tie than there will be a horizontal asymptote at the ratio of the leading coefficients.


Example Problem




What is the limit of the function as x approaches infinity?

The function has the same degrees on both top and bottum. You can use the infinity rule that the degree's of the function are a tie. Because they are a tie, this means that the limit is a horizontal asymptote at the ratio of the leading coefficients. In this case the ratio is 17/7.

ANSWER: 17/7


some extra sites on limits:

http://www.coolmath.com/limit1.htm
http://www.calculus-help.com/funstuff/phobe.html
http://curvebank.calstatela.edu/limit/limit.htm

Reminder:

hey nicole....you are up next!


Since it is almost halloween here is a cute joke:

Q: What do you get if you divide the cirucmference of a jack-o-lantern by its diameter?
A: Pumpkin Pi!