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§ 1.4

§ 1.4. Limits and the Derivative. Section Outline. Definition of the Limit Finding Limits Limit Theorems Using Limits to Calculate a Derivative Limits as x Increases Without Bound. Definition of the Limit. Finding Limits. EXAMPLE.

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§ 1.4

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  1. §1.4 Limits and the Derivative

  2. Section Outline • Definition of the Limit • Finding Limits • Limit Theorems • Using Limits to Calculate a Derivative • Limits as x Increases Without Bound

  3. Definition of the Limit

  4. Finding Limits EXAMPLE Determine whether the limit exists. If it does, compute it. SOLUTION Let us make a table of values of x approaching 4 and the corresponding values of x3 – 7. As x approaches 4, it appears that x3 – 7 approaches 57. In terms of our notation,

  5. Finding Limits EXAMPLE For the following function g(x), determine whether or not exists. If so, give the limit. SOLUTION We can see that as x gets closer and closer to 3, the values of g(x) get closer and closer to 2. This is true for values of x to both the right and the left of 3.

  6. Limit Theorems

  7. Finding Limits EXAMPLE Use the limit theorems to compute the following limit. SOLUTION Limit Theorem VI Limit Theorem II with r = ½ Limit Theorem IV

  8. Finding Limits CONTINUED Limit Theorems I and II Since , we have that:

  9. Limit Theorems

  10. Finding Limits EXAMPLE Compute the following limit. SOLUTION Since evaluating the denominator of the given function at x = 9 is 8 – 3(9) = -19 ≠ 0, we may use Limit Theorem VIII.

  11. Using Limits to Calculate a Derivative

  12. Using Limits to Calculate a Derivative EXAMPLE Use limits to compute the derivative for the function SOLUTION We must calculate

  13. Using Limits to Calculate a Derivative CONTINUED Now that replacing h with 0 will not cause the denominator to be equal to 0, we use Limit Theorem VIII.

  14. More Work With Derivatives and Limits EXAMPLE Match the limit with a derivative. Then find the limit by computing the derivative. SOLUTION The idea here is to identify the given limit as a derivative given by for a specific choice of f and x. Toward this end, let us rewrite the limit as follows. Now go back to . Take f(x) = 1/x and evaluate according to the limit definition of the derivative:

  15. More Work With Derivatives and Limits CONTINUED On the right side we have the desired limit; while on the left side can be computed using the power rule (where r = -1): Hence,

  16. Limits as x Increases Without Bound EXAMPLE Calculate the following limit. SOLUTION Both 10x + 100 and x2 – 30 increase without bound as x does. To determine the limit of their quotient, we employ an algebraic trick. Divide both numerator and denominator by x2 (since the highest power of x in either the numerator or the denominator is 2) to obtain As x increases without bound, 10/x approaches 0, 100/x2 approaches 0, and 30/x2 approaches 0. Therefore, as x increases without bound, 10/x + 100/x2 approaches 0 + 0 = 0 and 1 - 30/x2 approaches 1 – 0 = 1. Therefore,

  17. Limits as x Increases Without Bound CONTINUED

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