Introduction to limits Section 2.1 Introduction 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: lim 𝑥→𝑐 𝑓(𝑥) =𝐿. 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 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.

Extra Practice

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. 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?

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: 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.

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

Key Limits that are helpful to know

Sandwich Theorem