2. Aim: Evaluating Limits Course: Calculus Do Now: Aim: What are some techniques for evaluating limits? Sketch

3. Aim: Evaluating Limits Course: Calculus Do Now: Aim: What are some techniques for evaluating limits? If

4. Aim: Evaluating Limits Course: Calculus Do Now Graph: y = 3 y-int. – x-int. – Vertical asymptotes – Horizontal asymptotes – Plot several points x = 2 f(0)= 1/2 (0, 1/2) x – 2 = 0, x = 2 x-2145 y7/44/3-211/214/3 1/3 If degree of p = degree of q, then the line y = a n /b m is a horizontal asymptote. q(x) = 0 y = 3

5. Aim: Evaluating Limits Course: Calculus Asymptotes of Rational Functions Let f be the rational function given by where p(x) and q(x) have no common factors 1. The graph of f has vertical asymptotes at the zeros of q(x). 2.The graph of f has at most one horizontal asymptote, as follows: a) If degree of p degree of q, then the graph of f has no horizontal asymptote.

6. Aim: Evaluating Limits Course: Calculus 1.If p is a polynomial function and c is a real number, then 2.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 direct substitution Limits of Polynomial and Rational Functions

7. Aim: Evaluating Limits Course: Calculus Limits of Composite Functions If f and g are functions such that then = 4= 2 g(x)g(x) f(x)f(x) f(g(x)) = 2 4

8. Aim: Evaluating Limits Course: Calculus Limits of Trigonometric Functions Let c be a real number in the domain of the given trig function

9. Aim: Evaluating Limits Course: Calculus Evaluating Limits as x  a Finite Number c 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. 2.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

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

11. Aim: Evaluating Limits Course: Calculus Direct substitution results in an 0 in both numerator and denominator and will not yield a limit. Factor Out Dividing Out/Factoring Technique Find the

12. Aim: Evaluating Limits Course: Calculus Rationalize numerator Rationalizing Technique Find the Direct substitution results in an 0 in both numerator and denominator and will not yield a limit.

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

14. Aim: Evaluating Limits Course: Calculus Squeeze Theorem 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. Given: = 0 h(x)h(x)g(x)g(x) f(x)f(x) L =

15. Aim: Evaluating Limits Course: Calculus Special Trig Limits x is in radians

16. Aim: Evaluating Limits Course: Calculus More Special Trig Limits let y = 4x multiply top and bottom by 4

17. Aim: Evaluating Limits Course: Calculus Model Problems Evaluate: