Evaluating Limits Analytically Section 1.3

After this lesson, you will be able to: evaluate a limit using the properties of limits develop and use a strategy for finding limits evaluate a limit using dividing out and rationalizing techniques

Limits Analytically In 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.

Theorem 1.1 Some Basic Limits Let b and c be real numbers and let n be a positive integer. ____ ____ ____ Examples Let Let Let Think of it graphically y scale was adjusted to fit As x approaches 5, f(x) approaches 125 As x approaches 3, f(x) approaches 4 As x approaches 2, f(x) approaches 2

Direct Substitution Some 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.

Theorem 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: and Scalar multiple: Sum or difference: Product: Quotient: Power:

Limit 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.

Limit 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.

Theorem 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.

Theorem 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

Limit of a Composite Function-part a MAT 224 SPRING 2007 Limit of a Composite Function-part a Example: Given and , find a) First find Direct substitution works here

Limit of a Composite Function -part b b) Then find Direct substitution works here, too. ********************************************* Therefore,

Limits of Trig Functions If c is in the domain of the given trig function, then

Limits of Trig Functions Examples:

Limits of Trig Functions Examples:

Finding 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.

Example 1- Factoring Example* : Factor the denominator Example* : Direct substitution at this point will give you 0 in the denominator: Now direct substitution will work Graph on your calculator and use the table to check your result

Example 2- Factoring Example: Direct substitution results in the indeterminate form 0/0. Try factoring.

Example: Limit DNE Example*: Sum of cubes Not factorable Example*: 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.

Example 1- Rationalizing Technique First, we will try direct substitution: Indeterminate Form Plan: Rationalize the numerator to come up with a related function that is defined at x = 0.

Example 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 work Go ahead and graph to verify.

Example 2- Rationalizing Technique

Two Very Important Trig Limits (A star will indicate the need to memorize!!!)

Example 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 1 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.

Example 2 Try to get into the form:

Example 3

Example 4 Get into the form:

Example 5

Homework Section 1.3: page 67 #1, 5-39 odd, 49-61 odd, 67-77 odd