Section 1.2: Finding Limits Graphically and Numerically Day 1 AP Calculus AB

Math Talk Which is greater 24∙6 or 26∙4?

Presentation Key Colors RED: important and key to write down GREEN: important to write down, but you’ll need to summarize in own words. Also could be further example problems that you may or may not need. This is more up to your own discretion BLUE: do not have to write down at all. Just used to illustrate a concept we’ll be discussing or exploring an idea before we formalize it

By the end of this lesson, you should … Define components of limit notation Define a limit Determine the limit using a table Determine the limit using a graph Explain the difference between a limit and a function value Identify and explain where limits do not exist Draw graphs that show different limit/non-limit criteria

Limit Notation Example: “What is the limit of 2𝑥+4 as 𝑥 approaches −6?

Informal Limit Definition Example: lim 𝑥→1 𝑥 3 −1 𝑥−1 =3 “The limit of 𝑥 3 −1 𝑥−1 as x approaches 1 is 3.” Graphical: LIMIT Verbal: lim 𝑥→𝑐 𝑓(𝑥) =𝐿 If 𝑓(𝑥) becomes arbitrarily close to L as x approaches c from either side, then the limit as x approaches c is L. In your own words:

Graphical Approach Find lim 𝑥→−2 𝑓(𝑥) for the following functions using their respective graphs EX 1: EX 2: EX 3:

Graphical Approach EX 1: EX 2: EX 3: lim 𝑥→−2 𝑓 𝑥 =−1 lim 𝑥→−2 𝑓 𝑥 =−5 𝑓 −2 =1 𝑓 −2 =−5 𝑓 −2 =𝐸𝑟𝑟𝑜𝑟

Numerical Approach Find lim 𝑥→0 𝑓(𝑥) for the following functions using their respective tables EX 1: EX 2: EX 3: x -0.01 -0.001 -0.0001 0.0001 0.001 0.01 F(x) 1.99499 1.99950 1.99995 Error 2.00005 2.00050 2.00499 x -0.01 -0.001 -0.0001 0.0001 0.001 0.01 F(x) 14.9921 14.9960 14.9998 -7 15.0003 15.0056 15.0989 x -0.01 -0.001 -0.0001 0.0001 0.001 0.01 F(x) -3.0789 -3.0097 -3.0001 -3 -2.9996 -2.9987 -2.9214

Numerical Approach EX 1: EX 2: EX 3: lim 𝑥→0 𝑓 𝑥 =2 𝑓 0 =Error -0.01 -0.001 -0.0001 0.0001 0.001 0.01 F(x) 1.99499 1.99950 1.99995 Error 2.00005 2.00050 2.00499 lim 𝑥→0 𝑓 𝑥 =2 𝑓 0 =Error x -0.01 -0.001 -0.0001 0.0001 0.001 0.01 F(x) 14.9921 14.9960 14.9998 -7 15.0003 15.0056 15.0989 lim 𝑥→0 𝑓 𝑥 =15 𝑓 0 =−7 x -0.01 -0.001 -0.0001 0.0001 0.001 0.01 F(x) -3.0789 -3.0097 -3.0001 -3 -2.9996 -2.9987 -2.9214 lim 𝑥→0 𝑓 𝑥 =−3 𝑓 0 =−3

Calculator Shortcuts: Using a Table to Find the Limit Using the table to help find the limit of a function Example: lim 𝑥→4 𝑥 2 −16 = ? Step 1: Put 𝑓 𝑥 = 𝑥 2 −16 into “y =_____” Step 2: Click “2nd” “Window/Tblset” “Indpnt”: should be on “auto” “∆Tbl”: should be a very small change in x (ex: .0001) “TblStart”: near the value you are examining (ex: 4) Step 3: Click “2nd” “Graph/Table” Step 4: Analyze Results lim 𝑥→4 𝑥 2 −16 =0

Summary of Finding Limits Conceptually In your own words, take 5 minutes to explain the following for your notes 1. How to find the limit using a table Look at numbers VERY close to what value you are looking for on both sides 2. How to find the limit using a graph (this is really key!) Use a vertical line to cover up the value you are looking for Examine where both sides of the graph are approaching 3. What is the difference between a limit and a function value? How do they related to each other? Limit: what occurs around a particular x-value Function Value: where the function is defined at a particular x-value Function value does not affect the limit at that point

Limits that Fail to Exist Case 1: Behavior That Differs from Right and Left Example: lim 𝑥→0 𝑥 𝑥 Notice that the red lines are not meeting at one point Also, at x = 0 is the only place where the limit does not exist. At any other point in the function, the limit does exist.

Limits that Fail to Exist Case 2: Unbounded Behavior (some type of ±∞) Example: lim 𝑥→0 1 𝑥 2 Once again, only at x = 0, the limit does not exist. It does exist every where else. Key: ±∞ is not a value, thus a function cannot be bounded by it.

Limits that Fail to Exist Case 3: Oscillating Behavior Example: lim 𝑥→0 sin 1 𝑥 No matter how much you zoom in to x = 0, the function keeps bouncing between -1 and 1. Thus, we cannot conclude a limit.

Exit Ticket For Feedback Refer to the graph 𝑓(𝑥) 1. lim 𝑥→ −5 𝑓(𝑥) = 2. lim 𝑥→2 𝑓(𝑥) = 3. 𝑓 −5 = 4. At what values of x does the limit not exist? 𝑓(𝑥)