Infinite Limits and Limits to Infinity: Horizontal and Vertical Asymptotes

Recall… The notation tells us how the limit fails to exist by denoting the unbounded behavior of f(x) as x approaches c. Infinity is not a number!

Properties of Infinite Limits Let c and L be real numbers and let f and g be functions such that and 1.Sum or difference: Consider:

Properties of Infinite Limits Let c and L be real numbers and let f and g be functions such that and 1.Product: if L > 0 if L < 0 Consider:

Properties of Infinite Limits Let c and L be real numbers and let f and g be functions such that and 1.Quotient: Consider:

Definition - Vertical Asymptotes If f(x) approaches infinity (or negative infinity) as x approaches c from the left or the right, then the line x = c is a vertical asymptote of the graph of f. vertical asymptote

Determining Infinite Limits

The pattern… Is c even or odd? Sign of p(x) when x = c oddpositive oddnegative evenpositive evennegative and c is a positive integer

Using the pattern…

Limits at Infinity denotes that as x increases without bound, the function value approaches L L can have a numerical value, or the limit can be infinite if f(x) increases (decreases) without bound as x increases without bound

Horizontal Asymptotes The line y = L is a horizontal asymptote of f if or Notice that a function can have at most two HORIZONTAL asymptotes (Why?)

0 0 Horizontal Asymptote(s):__________

Note: It IS possible for a graph to cross its horizontal asymptote!!!!!! 2 2 Horizontal Asymptote(s):__________

0 1

0 0

Theorem – Limits at Infinity 1.If r is a positive rational number and c is any real number, then The second limit is valid only if x r is defined when x <

Using the Theorem

Guidelines for Finding Limits at ±∞ of Rational Functions 1.If the degree of the numerator is ___________ the degree of the denominator, then the limit of the rational function is ___. 2.If the degree of the numerator is _______ the degree of the denominator, then the limit of the rational function is the __________________ _______________________. 3.If the degree of the numerator is ___________ the degree of the denominator, then the limit of the rational function _______________. greater than 0 less than equal to the ratio of the leading coefficients is infinite

Using the Guidelines… ∞