Tuesday, October 10, 2006

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:


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


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:



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!


At 11:18 AM, Blogger nicole said...

I like your first link. I was absent on the day of the lesson, so I was very lost on the concept of limits. The geometric example was an odd but effective way to explain the basic definition of a limit. That website explains calculus concepts in a simple way which is also different from the way it is presented to us. It was very helpful!


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