2 6 Limits at Infinity Horizontal Asymptotes In

2. 6 Limits at Infinity; Horizontal Asymptotes: In this section, we let x become arbitrarily large (pos. or neg. ), and see what happens to f (x). What happens to the graph of f(x) as x goes to ∞ or -∞?

no y=0 a b

limits w/regard to vertical asymptotes: limits w/regards to horizontal asymptotes: EX #1: Find the infinite limits, limits at infinity, and asymptotes for the function f whose graph is shown below. 4 -1

Using previous properties, break the limit into parts. 1 Using horizontal asymptote properties. The leading powers are equal. HA y = 2/1

Using horizontal asymptote properties. Top power < Bottom power. HA y = 0 Using horizontal asymptote properties. The leading powers are equal. HA y = 2/3 Using horizontal asymptote properties. Top power > Bottom power. No HA

Divide the top and bottom by the x with the largest power from the bottom.

Rational ______functions will have _____ limit as x +∞ and x -∞ (H. A. ) SAME But what if NOT rational? ? ? . .

TWO Radical functions can have _________ horizontal asymptotes/limits!!

Think Conjugates

Polynomial NEVER Find Left and Right hand behaviors.

False statement…x must approach a constant. From Sect 2. 3.

EX #5 Evaluate.

Basic exponential function has a HA at y = 0.

= DNE Sine oscillates between 1 and -1 infinitely often, so we don’t approach any definite number.