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Aim: What are some techniques for evaluating limits?

Aim: What are some techniques for evaluating limits?. Do Now:. Sketch. Aim: What are some techniques for evaluating limits?. If. Do Now:. y -int. –. x -int. –. Vertical asymptotes –. Horizontal asymptotes –. y = 3. Plot several points. x = 2. Do Now. Graph:. f (0).

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Aim: What are some techniques for evaluating limits?

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  1. Aim: What are some techniques for evaluating limits? Do Now: Sketch

  2. Aim: What are some techniques for evaluating limits? If Do Now:

  3. y-int. – x-int. – Vertical asymptotes – Horizontal asymptotes – y = 3 Plot several points x = 2 Do Now Graph: f(0) = 1/2 (0, 1/2) 1/3 q(x) = 0 x – 2 = 0, x = 2 Ifdegree of p = degree of q, then the line y = an/bm is a horizontal asymptote. y = 3

  4. Let f be the rational function given by Asymptotes of Rational Functions where p(x) and q(x) have no common factors 1. The graph of f has vertical asymptotes at the zeros of q(x). The graph of f has at most one horizontal asymptote, as follows: a) If degree of p < degree of q, then the x-axis (y = 0) is a horizontal asymptote. b) If degree of p = degree of q, then the line y = an/bm is a horizontal asymptote. c) If degree of p > degree of q, then the graph of f has no horizontal asymptote.

  5. direct substitution Limits of Polynomial and Rational Functions • If p is a polynomial function and c is a real number, then • If r is a rational function given by r(x) = p(x)/q(x), and c is a real number such that q(c)  0, then

  6. If f and g are functions such that then Limits of Composite Functions = 4 = 2 4 f(x) g(x) = 2 f(g(x))

  7. Limits of Trigonometric Functions Let c be a real number in the domain of the given trig function

  8. To evaluate a limit algebraically as x approaches a finite number c, substitute c into the expression. 1. If the answer is a finite number, that number is the value of the limit. • If the answer is of the form 0/0, we have an indeterminate form. • Factor the numerator or denominator, simplify, substitute for c • Rationalize the numerator or the denominator, simplify, substitute • Simplify complex fraction, substitute Evaluating Limits as x a Finite Number c

  9. Find the Divide Out Dividing Out/Factoring Technique q(-3) = 0 Problem: direct substitution results in an indeterminate form. Note: this technique works only when direct substitution results in zeros in both numerator and denominator.

  10. Factor Out Dividing Out/Factoring Technique Find the Direct substitution results in an 0 in both numerator and denominator and will not yield a limit.

  11. Find the Rationalize numerator Rationalizing Technique Direct substitution results in an 0 in both numerator and denominator and will not yield a limit.

  12. = e Find the Using Technology Technique Table of values starting at –0.003 Graph Take average: (2.7196 + 2.7169)/2  2.71825 Use zoom and trace to find coordinates that are equidistant from x = 0 and take average of corresponding y’s.

  13. If h(x) <f(x) <g(x) for all x in an open interval containing, c, except possibly at c itself, and if then exists and is equal to L. Squeeze Theorem Given: h(x) g(x) L = = 0 = 0 f(x)

  14. Special Trig Limits x is in radians

  15. More Special Trig Limits multiply top and bottom by 4 let y = 4x

  16. Model Problems Evaluate:

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