Vertical Asymptotes VA If then xa is a VA of fx To find VA algebraically set denominator 0 Example 1 Find VA Finding limits on either side of a VA ID: 551092
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Slide1
3. Limits and infinitySlide2
Vertical Asymptotes (VA)
If
then
x=a is a VA of f(x)To find VA algebraically – set denominator = 0Slide3
Example 1 – Find VASlide4
Finding limits on either side of a VA
Plug in a value greater than VA and see if you get a positive or negative number
If positive then
If negative thenNow plug in a value less than VA and see what sign you getIf positive thenIf negative then If it is a one-sided limit, you only need to do one of theseSlide5
Example 2Slide6
Horizontal asymptotes (HA)
If or , the line y = b is an HA
A horizontal asymptote is NOT a discontinuity
A graph can cross its HA any number of times (although they don’t have to)An HA only describes what may happen to a function at its extreme endpoints (its end behavior)Slide7
Finding HA algebraically
Find the highest exponent in the numerator and denominator
If they are the same, divide coefficients to get HA
If the higher one is on bottom, y=0 is HAIf the higher one is on top, there is a slant asymptote (do long division to find it)When you take the limit as x approaches infinity or negative infinity for this third type, the answer will be either infinity or negative infinity. To find out which one, plug in a positive or negative value and see what type of answer you get.Slide8
Example
3
– find horizontal asymptoteSlide9
Example
4Slide10
Non-explicit degrees
Sometimes the degrees of the numerator and denominator are not explicit (like when they are under a radical)
Example 4
What about Slide11
Growth rates
Different functions grow at different rates for large values
Log functions grow slowly
Polynomials grow faster by order of degreeExponential functions grow faster than any polynomialFactorials are nextxx are the king of growthYou can find limits at infinity by analyzing respective growth limitsSlide12
Example
6Slide13
Limits that don’t involve infinity but are nonetheless special
Knowing the transformations of this and knowing where the jump occurs (the value that yields 0/0) will allow you to answer limit questions about this functionSlide14
Example 6
Evaluate the followingSlide15
Limits that don’t involve infinity but are nonetheless special
Piecewise functions
Example 8Slide16
Summary
If there is a limit as x approaches , there is a horizontal asymptote at the limit
If the limit equals , there is a vertical asymptote at the x value