2After this lesson, you will be able to: Evaluate a limit using the properties of limitsDevelop and use a strategy for finding limitsEvaluate a limit using dividing out and rationalizing techniquesEvaluate a limit using Squeeze Theorem
3Limits AnalyticallyIn the previous lesson, you learned how to find limits numerically and graphically. In this lesson you will be shown how to find them analytically…using algebra or calculus.
4Theorem 1.1 Some Basic Limits Let b and c be real numbers and let n be a positive integer.Examples 1:Think of it graphically…LetLetLet(y scale was adjusted to fit)As x approaches 3, f(x) approaches 4As x approaches 2, f(x) approaches 2As x approaches 5, f(x) approaches 125
5Direct SubstitutionSome limits can be evaluated by direct substituting for x.Direct substitution works on continuous functions.Continuous functions do not have any holes, breaks or gaps.Note: Direct substitution is valid for all polynomial functions and rational functions whose denominators are not zero (or not approaches to zero) as the x approaches to a certain value.ExampleHowever
6Theorem 1.2 Properties of Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:andScalar multiple:Sum or difference:Product:Quotient:Power:
7Theorem 1.2 Properties of Limits Note: The following assumptions are necessary to those properties of limits. You need provide all the counterexamples to some of the properties if the assumptions are not provided.andScalar multiple:Sum or difference:Product:Quotient:Power:
8Limit of a Polynomial Function Example 2:Since a polynomial function is a continuous function, then we know the limit from the right and left of any number will be the same. Thus, we may use direct substitution.Ans: The limit is 5
9Theorem 1.3 Limits of Polynomial and Rational Function If p is a polynomial function and c is a real number, thenIf p(x) and q(x) are polynomial functions and r(x) = p(x)/q(x) and c is a real number such that q(c) ≠ 0, then
10Limit of a Rational Function Make sure the denominator doesn’t = 0 !Example 3:Ans: 1If the denominator had been 0, we would NOT have been able to use direct substitution.
11Theorem 1.4 The Limit of a Function Involving a Radical Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c > 0 if n is even.
12Theorem 1.5 The Limit of a Composite Function If f and g are functions such that lim g(x) = L and lim f (x) = f (L), then
13Limit of a Composite Function-part a Example 4: Givena) FindDirect substitution works here
14Limit of a Composite Function -part b Example 4: Givenb) FindDirect substitution works here
15Limit of a Composite Function -part c Example 4: Givenc) FindFrom part a, we know that the limit of f(x) as x approaches 4 is 21
16Theorem 1.6 The Limits of Trig Functions Let c be a real number in the domain of the given trigonometric function22.214.171.124.5.6.
18A Strategy for Finding Limits Try Direct SubstitutionIf the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to divide out common factors or to rationalize the numerator or the denominator so that direct substitution works.Use a graph or table to reinforce your result.
19Theorem 1.7 Functions That Agree at All But One Point Let c be a real number and let f (x) = g (x) for all x ≠ c in an open interval containing c. If exists, thenalso exists and
20Example- Factoring Example 6: Direct substitution at this point will give you 0 in the denominator. Using a bit of algebra, we can try to find the limit.Now direct substitution will workGraph on your calculator and use the table to check your result
21Example- Factoring Example 7: Direct substitution results in 0 in the denominator. Try factoring.Now direct substitution will work.Use your calculator to reinforce your result.
22ExampleSum of cubesNot factorableExample 8:Direct substitution results in 0 in the denominator. Try factoring.None of the factors can be divided out, so direct substitution still won’t work.The limit DNE. Verify the result on your calculator.The limits from the right and left do not equal each other, thus the limit DNE.Observe how the right limit goes to off to positive infinity and the left limit goes to negative infinity.
23Example- Rationalizing Technique Direct substitution results in 0 in the denominator. I see a radical in the numerator. Let’s try rationalizing the numerator.Multiply the top and bottom by the conjugate of the numerator.Note: It was convenient NOT to distribute on the bottom, but you did need to FOIL on the topNow direct substitution will workGo ahead and graph to verify.
24Theorem 1.8 The Squeeze Theorem If h(x) ≤ f (x) ≤ g(x) for all x in an open interval containing c, except possibly at c itself, and ifthen
25Two Special Trigonometric Limits In your text, read about the Squeeze Theorem on page 65. Following the Squeeze Theorem are the proofs of two special trig limits…I will not expect you to be able to prove the two limits, so you’ll just want to memorize them. The next slide will give them to you and then we’ll use them in a few examples.
26Trig Limits (A star will indicate the need to memorize!!!) Think of this as the limit as “something” approaches 0 of the sine of “something” over the same “something” is equal to 1.(A star will indicate the need to memorize!!!)
27Example- Using Trig Limits =1Before you decide to even use a special trig limit, make sure that direct substitution won’t work. In this case, direct substitution really won’t work, so let’s try to get this to look like one of those special trig limits.This 5 is a constant and can be pulled out in front of the limit.Example 10, page 66, in your text is another example.Now, the 5x is like the heart. You will need the bottom to also be 5x in order to use the trig limit. So, multiply the top and bottom by 5. You won’t have changed the fraction. Watch how to do it.
28ExampleExample 12Direct substitution won’t work. We can use the sine trig limit, but first we’ll have to use some algebra since we need the bottom to be a 3x.To create a 3x on the bottom, we’ll multiply the bottom by 3/2. To be “fair”, we’ll have to multiply the top by 3/2 as well. Watch how I would do it.= 1
29Example Looking at the unit circle, this value is 1 Example 13 This example was thrown in to keep you on your toes. Direct substitution works at this point since the bottom of the fraction will not be 0 when you use π/2.
30ExampleExample 14This is the 2nd special trig limit and you should know that this limit is 0. Let’s prove it by using the 1st trig limit.Use the Pythagorean Trig IdentityAgain, in this case, it’s best not to distribute on the bottom. You’ll see why it helps to leave it in factored form for now.Be creativeSpecial Trig Limit