Section 2.1INTRODUCTION TO LIMITS

Definition of a Limit

Limits allow us to describe how the outputs of a function (usually the y or f(x) values) behave as the inputs (x values) approach a particular value.

Limit Notation

The previous definition is confusing, but all it really says is that a function has a limit at a particular value if the function doesn’t go crazy in the vicinity of that value.

In other words, if you only look at the x-values near the value you are trying to find a limit for, can you graph all of the nearby y-values in a window?

If you can, then we use the notation: . This means that the function approaches L as x

approaches c.

One and Two Sided Limits

When we say that the function “approaches” a particular value, it can do so moving from the left, or from the right.

Another way to think of limits

A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand limits at c exist and are equal. In other words the function must be approaching the same value from both sides.

Example

Graph the following function in your calculator.

Compare the limits and the values of the function at various spots on the graph.

Do-Now

Greatest Integer Function (Int x): The function for which…..Input: all real numbers x.Output: The largest integer less than or

equal to x. Sketch a graph for this function and complete

pg 63 #37-40.

Finding limits algebraically

Graph the following functions in your calculator.

1. 2.

What do the graphs of these functions tell you about the limit of the function as x approaches 3?

Limits of Rational Functions

Can you find the limit as x approaches 3 by using direct substitution?Why or why not?

Why did the limit not exist in #1 but it did in function #2?

Use algebra to simplify the expressions and confirm the limits that you found graphically.

Properties of Limits

Properties of Limits Continued

Calculator exercise:

Find: Find: Knowing these limits can allow you

to find other limits algebraically.

Examples

1. 2. 3. 4. (hint: change tan x to something

else)

14 2014 – APSI – Day 1

Key Limits that are helpful to know

1lim 0x x

0

1limx x

0

1limx x

0

sinlim 1x

xx

0

1 coslim 0x

xx

1

0lim 1 x

xx e

1lim 1

x

xe

x

0

1lim ( ) limx x

f x fx

Sandwich Theorem