Limits & infinity (horizontal & vertical asymptotes) AP Calc
16
The Lame Joke of the day.. What do cats eat for breakfast? And now it’s time for.. Mice Krispies!
Transcript
1. The Lame Joke of the day.. What do cats eat for breakfast?
And now its time for.. Mice Krispies!
2. Vertical Asymptotes Vertical asymptotes occur when a
function is undefined (usually the zeros of the denom.) At a
vertical asymptotes, the limit of the function is +/- infinity (
does not exist) At the asymptote, the function increases or
decreases without bound. Also, )(lim)(lim xfxf cxcx
3. 2 vertical asymptotes
4. Practice Find the vertical asymptotes of each function: 1 1
)( x xf 3 )2( 4 )( x xf 16 2 )( 4 2 t tt xf
5. Vertical Asymptotes - exceptions The exception is when you
can cancel common factors - YOU get a hole instead of an
asymptote
6. Limits at infinity How do you find the limit of a function
when x approaches +/- infinity? YOU FIND THE HORIZONTAL ASYMPTOTES
Tells you the end behavior of the graph.)(lim xf x
7. How do we find the horizontal asymptotes? Compare the
degrees of the numerator and denominator. Degrees equal: Its the
ratio of the leading coefficients. Degree of numerator is greatest:
No asymptote. The function is unbounded at the ends. (Goes on
forever to infinity) Degree of denominator is greatest: y = 0
(x-axis) is horizontal asymptote
8. Example 1 12 lim x x x
9. Example 13 52 lim 2 x x x
10. Example 13 52 lim 2 3 x x x
11. Practice Evaluate each limit xx 1 lim 13 23 lim 2 x x x 13
25 lim 4 4 x xx x