1. Pretty cheap training © 2008, Bob Wilder Limits and Continuity “Intuition” Limits Either Existor not ► If they exist, … you may know what they are … you may not ► If they do NOT exist, … you may know that … you may not!

2. Pretty cheap training © 2008, Bob Wilder Limits and Continuity No Limit  Limits don’t exist ► If there is a ‘jump discontinuity’ (pit?) … left sided limit and right sided limit unequal ► If there is a vertical asymptote (wall) … unbounded behavior ► If there is an oscillation (boing boing)... doesn’t get close to anything (and stay there) ► If end behavior tends to (zoom!) … unbounded behavior ► Note that removable discontinuities have limits … everywhere!

3. Pretty cheap training © 2008, Bob Wilder Limits and Continuity No Limit  Limits don’t exist ► If there is a ‘jump discontinuity’

4. Pretty cheap training © 2008, Bob Wilder Limits and Continuity No Limit  Limits don’t exist ► If there is a vertical asymptote (wall)

5. Pretty cheap training © 2008, Bob Wilder Limits and Continuity No Limit  Limits don’t exist ► If there is an oscillation (boing boing)

6. Pretty cheap training © 2008, Bob Wilder Limits and Continuity No Limit  Limits don’t exist ► If end behavior tends to (zoom!)

7. Pretty cheap training © 2008, Bob Wilder Limits and Continuity No Limit  Limits don’t exist ► Note that removable discontinuities have limits

8. Pretty cheap training © 2008, Bob Wilder Limits and Continuity Limits ! Limits that exist ► Discovery phase … graphically … from a table … any others? ► Verification phase … calculate … sometimes it’s just difficult

9. Pretty cheap training © 2008, Bob Wilder Limits and Continuity Limits ! Calculating limits ► Basic rules

10. Pretty cheap training © 2008, Bob Wilder Limits and Continuity Limits ! What you need to know – AP AB ► Substitute! ► Rational Functions - Factor! (eliminate 0/0) - Divide by highest degree x term ► Piecewise Functions (left side vs. right side) ► Radicals - rationalize - multiply by ‘conjugate’ ► “Rational Radicals to infinity” watch |x| and signs ► Trig identities

11. Pretty cheap training © 2008, Bob Wilder Limits and Continuity Limits ! Limits – things to keep in mind ► … polynomials taught us everything we feel is “right” ► except for the (*&%$& denominator, you could say the same about rational functions ► One-sided vs. two-sided limits ► Limits at infinity (end behavior) ► Remember so don’t be afraid ► Do not manipulate infinity like it’s a number Be respectful of the subtleties

12. Pretty cheap training © 2008, Bob Wilder Limits and Continuity Limits ! Limits – how to calculate them … ► Substitute! for every continuous function f (whatever that means) ► Rational Functions * numerator, denominator not 0 => see substitute! * numerator not 0; denominator 0 => unbounded * numerator AND denominator 0 => try to factor and cancel

13. Pretty cheap training © 2008, Bob Wilder Limits and Continuity Limits ! Limits – how to calculate them … ► Piecewise Functions does limit from right = limit from left? ► Radicals The most common problem is of the form infinity – infinity Try – Multiply top and bottom by conjugate Create a fraction first if necessary

14. Pretty cheap training © 2008, Bob Wilder Limits and Continuity Limits ! Limits – how to calculate them … ► Rational Functions to Infinity another technique is to divide the top and bottom by ‘variable to the highest degree’ idea: all but 1 or 2 pieces go to zero! ► Rational Radicals to Infinity (bob-ism) Sometimes conjugates help Usually you incorporate dividing by same thing BUT! Remember that you leave the sign outside when you bring things under the radical sign Lim to negative infinity

15. Pretty cheap training © 2008, Bob Wilder Limits and Continuity The nice property Continuity! ► Definition is critically important (5% of AP test) A function f is continuous at point x=a iff => exists  f is defined at x=a  = f(a) Need all 3 conditions (why?)

16. Pretty cheap training © 2008, Bob Wilder Limits and Continuity Continuity! ► Rigorous definition of limit (and continuity) was an extremely important development ► From what I remember … The difficult problems are functions defined piecewise AND you need to find the value of k that makes function continuous

17. Pretty cheap training © 2008, Bob Wilder Limits and Continuity Trigonometric stuff ► You gotta know ► x is in radians ► it follows that ► You gotta know sin 2 + cos 2 = 1