Introduction to Limits. What is a limit? A Geometric Example Look at a polygon inscribed in a circle As the number of sides of the polygon increases,

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Presentation transcript:

Introduction to Limits

What is a limit?

A Geometric Example Look at a polygon inscribed in a circle As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle.

If we refer to the polygon as an n-gon, where n is the number of sides we can make some mathematical statements: As n gets larger, the n-gon gets closer to being a circle As n approaches infinity, the n-gon approaches the circle The limit of the n-gon, as n goes to infinity is the circle

The symbolic statement is: The n-gon never really gets to be the circle, but it gets close - really, really close, and for all practical purposes, it may as well be the circle. That is what limits are all about!

FYI WAY Archimedes used this method WAY before calculus to find the area of a circle.

An Informal Description If f(x) becomes arbitrarily close to a single number L as x approaches c from either side, the limit for f(x) as x approaches c, is L. This limit is written as

Numerical Examples

Numerical Example 1 Let’s look at a sequence whose n th term is given by: What will the sequence look like? ½, 2/3, ¾, 5/6, ….99/100, 99999/100000…

What is happening to the terms of the sequence? Will they ever get to 1? ½, 2/3, ¾, 5/6, ….99/100, 99999/100000…

Let’s look at the sequence whose n th term is given by 1, ½, 1/3, ¼, …..1/10000, 1/ …… As n is getting bigger, what are these terms approaching ? Numerical Example 2

Graphical Examples

Graphical Example 1 As x gets really, really big, what is happening to the height, f(x)?

As x gets really, really small, what is happening to the height, f(x)? Does the height, or f(x) ever get to 0?

Graphical Example 2 As x gets really, really close to 2, what is happening to the height, f(x)?

Find Graphical Example 3

Use your graphing calculator to graph the following: Graphical Example 3 Find As x gets closer and closer to 2, what is the value of f(x) getting closer to?

Does the function exist when x = 2?

ZOOM Decimal

Limits that Fail to Exist

What happens as x approaches zero? The limit as x approaches zero does not exist. Nonexistence Example 1: Behavior that Differs from the Right and Left

Nonexistence Example 2 Discuss the existence of the limit

Nonexistence Example 3: Unbounded Behavior Discuss the existence of the limit

Nonexistence Example 4: Oscillating Behavior Discuss the existence of the limit X2/π2/3π2/5π2/7π2/9π2/11πX 0 Sin(1/x)11 1 Limit does not exist

Common Types of Behavior Associated with Nonexistence of a Limit

Definition of Limit: If lim x  c+ f(x) = lim x  c- f(x) = L then, lim x  c f(x)=L (Again, L must be a fixed, finite number.) f(2) = f(4) = Examples:

f(4) = f(0) =

f(6) = f(3) =