3. Limits and infinity

Vertical Asymptotes (VA) If then x=a is a VA of f(x) To find VA algebraically – set denominator = 0

Example 1 – Find VA

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 then Now plug in a value less than VA and see what sign you get If it is a one-sided limit, you only need to do one of these

Example 2

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)

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 HA If 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.

Example 3 – find horizontal asymptote

Example 4

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

Growth rates Different functions grow at different rates for large values Log functions grow slowly Polynomials grow faster by order of degree Exponential functions grow faster than any polynomial Factorials are next xx are the king of growth You can find limits at infinity by analyzing respective growth limits

Example 6

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 function

Example 6 Evaluate the following

Limits that don’t involve infinity but are nonetheless special Piecewise functions Example 8

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