1. Introduction to limits
Section 2.1
Introduction to limits

2. 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.

3. 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:
lim 𝑥→𝑐 𝑓(𝑥) =𝐿. This means that the function approaches L as x approaches c.

4. 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.

5. 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.

6. Example 𝑓 𝑥 = 𝑥 2 −2, 𝑥 ≤0 𝑓 𝑥 = 𝑥 2 −2𝑥 −8 𝑥 −4 , 𝑥>0
Graph the following function in your calculator.
𝑓 𝑥 = 𝑥 2 −2, 𝑥 ≤0 𝑓 𝑥 = 𝑥 2 −2𝑥 −8 𝑥 −4 , 𝑥>0
Compare the limits and the values of the function at various spots on the graph.

7. Extra Practice

8. 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.

9. Finding limits algebraically
Graph the following functions in your calculator.
1. 𝑓 𝑥 = 𝑥 2 −6𝑥+2 𝑥 −3
2. 𝑓 𝑥 = 𝑥 2 −5𝑥+6 𝑥 −3
What do the graphs of these functions tell you about the limit of the function as x approaches 3?

10. 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.

11. Properties of Limits

12. Properties of Limits Continued

13. Calculator exercise: Find: lim 𝑥 →0 sin 𝑥 𝑥 Find: lim 𝑥 →0 1 − cos 𝑥 𝑥
Knowing these limits can allow you to find other limits algebraically.

14. Examples 1. lim 𝑥 →0 tan 𝑥 𝑥 2. lim 𝑥 →0 1 − 𝑐𝑜𝑠 2 𝑥 𝑥
4. lim 𝑥 →𝜋/ − tan 𝑥 sin 𝑥 − cos 𝑥 (hint: change tan x to something else)

15. Key Limits that are helpful to know

16. Sandwich Theorem