Limits in Calculus JON PIPERATO CALCULUS GRADES 11-12.

Limits in Calculus JON PIPERATO CALCULUS GRADES 11-12

Limits in Calculus, an Introduction Limits are a major component to Calculus, which is used everyday Integrals take the limit and use it to find an area under a curve Finding surface area, volume, distances etc. Electric charge All engineering and all science

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Main Menu Lesson 1: Intuitive Definition and Graphing Lesson 2: Limits in Tables Lesson 3: Limits that fail to exist Lesson 4: Algebraic Limits Lesson 5: Piecewise Functions Lesson 6: Infinite Limits Lesson 7: Discontinuity **Clicking the information button will take you to the beginning of each lesson **? Button takes you to review problems

Lesson 1: Intuitive Definition & Graphing A limit of a function is described as the behavior of that function as it approaches a specific point EX: This reads: as x approaches c, the function f(x) approaches the real number L. In other words, as x gets closer to c, that f(x) gets closer to LTHIS DOES NOT MEAN f(c)=L,

Limits in Graphing lim f(x)= 2 X 4 lim f(x)= 4 x-1 lim f(x)= -2 X -4 + lim f(x)= 3 X -4 - lim f(x)= DNE X -4 lim f(x)= -1 X 2 - lim f(x)= -1 X 2 + f(2)= 3

For limits approaching an x- value with two points, if a direction is not specified, the limit DNE Ex. lim f(x)= DNE X -4 If a direction is specified, use you finger and follow the line from that direction until you reach the point. Ex. lim f(x)= -2 X -4 + If a specific function is given, you are looking for the point that is not connect to the lines in the graph Ex. f(2) = 3 Limits in Graphing Continued

Examples of Intuitive Definition Write the following limits in sentence form: 1) 2) 3) Click here to see answers!

Answers to Intuitive Definition 1)the limit of f(x) as x approaches a is L 2) the limit of 1/x as x approaches infinity is 0 3)the limit as 1/x as x approaches 0 is infinity

Limit Graphing Example lim f(x)= X -4 lim f(x)= X -2 lim f(x)= X 1 lim f(x)= X 1 - lim f(x)= X 1 + f(1)= f(-2)= Click here to see the answers!

Graphing Limits Answers lim f(x)= X -4 lim f(x)= X -2 lim f(x)= X 1 lim f(x)= X 1 - lim f(x)= X 1 + f(1)= f(-2)= -2 3 DNE -3 4 2 5

You Have Completed Lesson 1! Summary Limits are defined as the characteristics of the function as it approaches a specific point. Intuitive Definition is writing a limit in sentence form Graphing limits can be used to find a specific location of functions as they approach a specific point on a graph

Lesson 2: Limits in Tables There are multiple ways to solve for limits in Calculus This is a second option to solving limits For this lesson, you will need a TI-84 calculator to complete the tasks

Limits in Tables The first step of solving limits through tables is to identify what variable x is going to in the limit lim f(x 2 -9/x+3)= x -3 Next, knowing that x is approaching -3, you must construct your table using numbers close to the variable in the statement above. Numbers such as -3.1, 3.01, 3.001 & -2.9, -2.99, -2.999 Then put the equation into the calculator Finally use the calculator to find the missing values and estimate what the pattern of the table is Dont panic, we will go through problems together

Limits in Tables Here is an example of what the table from the previous problem would look like -3.1-6.1 -3.01-6.01 -3.001-6.001 -3 -2.999-5.999 -2.99-5.99 -2.9-5.9 Once the numbers are found, you must use the pattern to figure out was - 3 is equal toit is equal to -6 Therefore lim f(x)= -6 x-3

Limit in Tables Examples Try these limits with a partner: Click here to see the answers!

Limits in Tables Answers = 40 = 1/2 = 6

You Have Completed Lesson 2 Summary In conclusion we discovered a way to solve for a limit using our calculators and tables Without the proper calculator this problem will be a pain, so please see me if you must borrow one!

Lesson 3: Limits That Fail to Exist A limit doesn't exist if the function is not continuous at that point. To check if a limit exists or not, graphically, you must approach it from the left and right side and if they are not equal, they do not exist. One type of limit that fails to exist is a jump. A jump can be found in the graphing of the function |x|/ x Another is a vertical asymptote A vertical asymptote is when x=0 Lastly, f(x) oscillates between two fixed values as x approaches c

Examples Of Discontinuities Match the following discontinuities with the possible graph |x|/ x 1/x 2 Sin (1/x) Click here to see the answers!

Answer to Discontinuities |x|/ x Sin (1/x) 1/x 2

You Have Completed the Lesson!

Lesson 4: Algebraic Limits We can solve certain limits with our knowledge of algebra! All we have to do is plug the number x is going towards in the equation But of course there is a catch! You cannot just simply plug that number in if it makes the equation false. We cannot make the denominator zero

Algebraic Limits An example of the information on the previous slide is: Since plugging -2 in will give us zero in the denominator, we must do some solvingwith algebra!!!! However, if it plugs in without a problem, then you just solve!

Algebraic Limits There are three ways to solve for a limit 1) Limits by direct substitution 2) Limits by FactoringYay! 3) Limits by Rationalization A video on the next slide will explain how to solve the following limits

Algebraic Limits Video

Algebraic Limits Limits by direct substitution Exactly how it sounds Just plug the number in! Limits through factoring Use factoring to cross out unwanted denominator Limits through Rationalization Use the reciprocal to get rid of the unwanted denominator

Algebraic Limit Example: Select one to be your answer: A) 10 B) 5C) 2D) 2 A) 10 B) 5C) 2D) 2

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Algebraic Limits Get with a partner, and solve the following limits algebraically through factoring 1) 2) 3) The button for the answers will be on the next page

Algebraic Limits Stay with your partner and work on the following rationalization problems 1) 2) Click here to see the answers!

Algebraic Limits Answers = 11/4 = 4 = 2 = 1/4 = 1/2

You Have Finished Lesson 4: Algebraic Limits Summary In summary there are three different ways to solve for an algebraic limit Direct substitution Factoring Rationalization Watch the video before coming to class to get a better understanding of the lesson!

Lesson 5 Piecewise Functions A piecewise function look like this : We already have the skills from previous sections to solve this problem so do not let looks deceive you! It is called piecewise because it is broken into pieces on a graph, but it is no different then solving ordinary functions!

Limits of a Piecewise Function Example: Solve for f(5) Use the function that satisfies the number five Clearly, you must use the second since 5>0 You then just you substitution from last section (5) 2 = 25

Limits of a Piecewise Function Lets try one with a limit lim f(x) = x -2 - So first we must determine which piece to use Then use direct substitution We can determine that we must use the first piece since x is less than negative 2 (coming from the left) Using direct substitution we plug in -22(-2)+8= -4+8= 4 Therefore lim f(x) = x-2 - 4

Limits of a Piecewise Function Review Time for you to try some on your own! lim f(x) = x5 Will the answer be A) 10 A) 10 B)5 B)5 C) 0 C) 0

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One more example just to be sure! lim f(x) = x0 What is your answer? A) 4 A) 4 B) 8 B) 8 C) 5 C) 5 Limits of a Piecewise Function

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You Have Finished Lesson 5: Piecewise Functions Summary Solving for a piecewise function is easy for us, because we already learned the process! The only difference is that the function we are looking at is broken up into different parts We will look at why they are broken up in an upcoming lesson Any questions?

Lesson 6 Infinite Limits These problems can be extremely simpleas long as you learn the three rules!!!!! The first rule: If the powers are the same (of the variable) in the denominator and numerator, then your answer will be the coefficients of the variables. The second rule: If the power of the variable in the numerator is higher then the denominator, then your answer is (infinite) The third rule: If the power of the variable in the numerator is less than the denominator, your answer is 0

Infinite Limits Refer to this table if you get confused

Infinite Limits Lets try some examples! First lets find the variables and what power they are to Since they are to the same power, we must you rule number one, which is to take their coefficient. So the coefficient of x is one and the coefficient of 3x is 3. Therefore your answer would be 1/3

Infinite Limits Example: Looking at the variables, we can conclude that they have different powers, and the denominator is bigger. Therefore we must use rule number three. With our knowledge we know that rule three makes our answer 0

Infinite Limits One last example! We can conclude that the numerators power is greater than the denominator, which means we will use the second rule. But wait!!!!! do not ignore the negative sign in front of the infinity symbol In this case the negatives will cancel out giving you a positive

Infinite Limits Examples Try some of these examples! 1) 2) 3) 4)

Infinite Limit Answers = 1/2 = 0 = - = 1/3

You Have Completed Lesson 6!

Lesson 7 Asymptote Graphs (Discontinuities) Remember in lesson 5, piecewise functions, I said we would look at why they were in pieces? There are three different reasons for asymptotes in a graph that will be touched on in the lesson JumpsPiecewise functions A limit at infinity (1/x or 1/x 2 ) A removable discontinuity

Discontinuities Jump

Discontinuities Limit at infinity

Discontinuities Removable

Lesson 7 Discontinuity We talked about the three types of discontinuities and examples of each Removable Jumps Limits at infinity Which of the three types are scene in the graphing approach of solving limits? A) RemovableRemovable B)JumpsJumps C) Limits at InfinityLimits at Infinity

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Limits in Calculus JON PIPERATO CALCULUS GRADES 11-12.

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