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 techniques
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.____________ExamplesLetLetLetThink of it graphicallyy scale was adjusted to fitAs x approaches 5, f(x) approaches 125As x approaches 3, f(x) approaches 4As x approaches 2, f(x) approaches 2
5Direct SubstitutionSome limits can be evaluated by direct substitution 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.
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:
7Limit of a Polynomial Function Example: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.
8Limit of a Rational Function Make sure the denominator doesn’t = 0 !Example:If the denominator had been 0, we would not have been able to use direct substitution.
9Theorem 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.So we can use direct substitution again, as long as c is in the domain of the radical function.
10Theorem 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
11Limit of a Composite Function-part a MAT SPRING 2007Limit of a Composite Function-part aExample: Given and ,finda) First findDirect substitution works here
12Limit of a Composite Function -part b b) Then findDirect substitution works here, too.*********************************************Therefore,
13Limits of Trig Functions If c is in the domain of the given trig function, then
16Finding Limits Try Direct Substitution If 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 so that direct substitution works.Use a graph or table to reinforce your result.
17Example 1- Factoring Example* : Factor the denominatorExample* :Direct substitution at this point will give you 0 in the denominator:Now direct substitution will workGraph on your calculator and use the table to check your result
18Example 2- Factoring Example: Direct substitution results in the indeterminate form 0/0. Try factoring.
19Example: Limit DNE Example*: Sum of cubesNot factorableExample*:Direct substitution results in 0 in the denominator. Try factoring.The limit DNE. Verify the result on your calculator.None of the factors can be divided out, so direct substitution still won’t work.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.
20Example 1- Rationalizing Technique First, we will try direct substitution:Indeterminate FormPlan: Rationalize the numerator to come up with a related function that is defined at x = 0.
21Example 1- Rationalizing Technique Multiply the numerator and denominator by the conjugate of the numerator.Note: It was convenient NOT to distribute in the denominator, but you did need to FOIL in the numerator.Now direct substitution will workGo ahead and graph to verify.
23Two Very Important Trig Limits (A star will indicate the need to memorize!!!)
24Example 1- Using Trig Limits This 5 is a constant and can be pulled out in front of the limit.Example*:Before you decide to even use a special trig limit, make sure that direct substitution won’t work. In this case, direct substitution won’t work, so let’s try to get this to look like one of those special trig limits.Note:Equals 1Now, 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.