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1 Evaluating Limits Analytically Section 1.3

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2 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

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3 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.

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

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5 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.

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

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7 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.

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8 Limit of a Rational Function Example: Make sure the denominator doesnt = 0 ! If the denominator had been 0, we would not have been able to use direct substitution.

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9 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.

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10 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

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11 Limit of a Composite Function- part a Example: Given and, a) First find Direct substitution works here find

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

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

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14 Limits of Trig Functions Examples:

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15 Limits of Trig Functions Examples:

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16 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.

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

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

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19 Example: Limit DNE Example*: Direct substitution results in 0 in the denominator. Try factoring. Sum of cubes Not factorable None of the factors can be divided out, so direct substitution still wont 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.

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20 Example 1- Rationalizing Technique Example*: 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.

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21 Example 1- Rationalizing Technique Example*: 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.

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22 Example 2- Rationalizing Technique Example:

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

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24 Example 1- Using Trig Limits Example*: Before you decide to even use a special trig limit, make sure that direct substitution wont work. In this case, direct substitution wont work, so lets try to get this to look like one of those special trig limits. 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 wont have changed the fraction. Watch how to do it. Note: Equals 1 This 5 is a constant and can be pulled out in front of the limit.

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25 Example 2 Try to get into the form:

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26 Example 3

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27 Example 4 Get into the form:

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28 Example 5

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29 Homework Section 1.3: page 67 #1, 5-39 odd, odd, odd

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