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Determining Limit Values

Determining Limit Values. Limits give us a language for describing how the outputs ( y values) of a function behave as the inputs ( x values ) approach some particular value . Sometimes we use direct substitution, factoring, rationalizing or expanding to calculate a limit.

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Determining Limit Values

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  1. Determining Limit Values • Limits give us a language for describing how the outputs (y values) of a function behave as the inputs (xvalues) approachsome particular value. • Sometimes we use direct substitution, factoring, rationalizing or expanding to calculate a limit.

  2. Definition of a Limit We write limf(x) = L x→a and say “the limit of f(x), as xapproachesa, equals L” if we can make the values off(x) arbitrarily close to L (as close to Las we like) by taking xto be sufficiently close to a, but not equal to a.

  3. Functions Continuous at ‘a’ A function is continuous when its graph is a single unbroken curve that you could draw without lifting your pen from the paper. That is not a formal definition, but it helps you understand the idea.

  4. Functions Continuous at ‘a’ Polynomial Functions are Continuousat all x values (1, 5) (1 3) 5 3 lim5 = x→1 5 lim (– 2x + 5) = x→3 3

  5. Functions Continuous at ‘a’ lim() = x→3

  6. Functions Continuous at ‘a’ 0 0 limf(x) = x→2 (2, 0)

  7. If direct substitution can be done without an error we can determine the function value and limit this way. lim(7 – 2x) = x→3 7 – 2(3) = 1 7 – 2(3) = 1 42 + 2(4) = 24 lim(x2 + 2x) = x→4 42 + 2(4) = 24 lim(2x2 – 3x + 4) = x→2 2(2)2– 3(2) + 4 = 6 2(2)2– 3(2) + 4 = 6

  8. Limits Rational Expressions If ris a rational function given by and c is a real number such that q(c) ≠ 0, then

  9. Let’s look at some examples to clarify this MATH LANGUAGE

  10. Remember: Division by 0 is undefined. Rational Functions are continuous at any x value that does not result in division by zero.

  11. Try it! By Substitution The function is continuous at x = 3

  12. So what is not continuous (also called discontinuous)? Look out for holes, jumps or vertical asymptotes.

  13. Exploring a Nonexistent Limit. Rational Functions are discontinuous at any x value that DOES result in division by zero. undefined There will be a vertical asymptote when both function and limit do not exist

  14. Exploring a Nonexistent Limit. x = 1 Use a table to show the limit does not exist. Find • Use a graph to show the • limit does not exist. The y values are getting further apart the closer x gets to 1 Since f (1) does not exist and the limit as x→1 does not exist there is a vertical asymptote at x = 1

  15. Finding Limits Algebraically when the Function is Undefined Direct substitution won’t WORK!!! DNE Factor and reduce There is a vertical asymptote f(5) does not exist Substitute – 5 in forx Undefined!

  16. Finding Limits Algebraically when the Function is Undefined Direct substitution won’t WORK!!! undefined HOWEVER as x→5

  17. Finding Limits Algebraically when the Function is Undefined Factor and reduce f(5) does not exist There is a “hole” at Substitute 5 in for x

  18. Lesson #5 WorksheetEXAMPLE 1: The function DOES NOT exist at ‘a’. -∞ Approaching 0.1666.. ∞

  19. Lesson #5 Worksheet EXAMPLE 1: Continued The function f(x) does not exist at x= +3 but the limits of f(x) as x approaches are a different matter. (-3, 0.166666) x = 3 undefined

  20. Limit of a Piecewise Function The function g has a limit of 2 as x →1 even though g(1) ≠ 2 (1, 2) (1, 1) (1, 1) The function is discontinuous when x = 1. There is a “hole” at (1, 2).

  21. YES! Limit of a Function Involving a Radical Is there a radical rule? 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.

  22. n is odd and > 0 .44

  23. Finding Limits By Graphing By Substitution (3, 2) 2 3

  24. Use a graph of to show that the following limit does not exist. Exploring a Nonexistent Limits There is no point on the graph wherex= – 3 By substitution: is a non-real number

  25. Remember This! Multiply by its conjugate 3 NOTE: The answer is a rational number

  26. Rationalizing The Numerator Rational number

  27. Find the following Limit by Rationalizing The Numerator Direct substitution won’t WORK!!! 1

  28. Lesson 5 continued—Example 3 Direct substitution won’t WORK!!! Multiply Top and Bottom by the conjugate. 1

  29. Lesson 5 Worksheet continued Example 2 Page 2 Practice Rationalizing: 1

  30. EXAMPLE 3: Expanding Technique Again substitution won’t WORK. Find

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