Limits at Infinity: End behavior of a Function

Limits of Polynomials as x approaches infinity or negative infinity The term with the highest power will determine the limit 1) 2)

Laws for Limits at Infinity

Limits of Rational Functions One technique for finding the end behavior of a rational is to divide each term in numerator and denominator by highest power of x in the denominator 1)

But easier than that…. Since a horizontal asymptote describes the end behavior of a function the limit as x approaches infinity or negative infinity is generally the same value for f(x) as the horizontal asymptote A quick review… If the degree of p(x) < the degree of q(x), there is a horizontal asymptote at y = 0 If the degree of p(x) > the degree of q(x), there is NO horizontal asymptote If the degree of p(x) = the degree of q(x), there is a horizontal asymptote at

Examples Would it have worked to find the HA for Example 1? 1) 2)

If no horizontal asymptote exists… The function either increases or decreases without bound because there is no HA to cause the graph to “level off” 1) 2)

Limits involving Radicals The limit of an nth root is the nth root of the limit 1)

Limits with Trig Functions Why would it be accurate to say that most trig. functions fail to have limits as x approaches infinity or negative infinity? (Hint: Think of the graphs)

Limits of and Consider the graphs of

Pg. 130 (1 – 17 odd, 25, 27)