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