Functions with Asymptotes and Critical Points (8.5.5)

Question 1

• Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero. A vertical asymptote to the graph of a function f is a line whose equation is x = a, where lim x -> a+ f(x) = ±∞, or lim x-> a- f(x) = ±∞.

Answer 1

• Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero. A vertical asymptote to the graph of a function f is a line whose equation is x = a, where lim x -> a+ f(x) = ±∞, or lim x-> a- f(x) = ±∞.

Question 2

• Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity. A horizontal asymptote to the graph of a function f is a line whose equation is y=a, where lim x -> ∞ f(x) = a, or lim x->-∞ f(x) = a.

Answer 2

• Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity. A horizontal asymptote to the graph of a function f is a line whose equation is y=a, where lim x -> ∞ f(x) = a, or lim x->-∞ f(x) = a.

Question 3

• The behavior of a function can change from one side of a vertical asymptote to the other.

Answer 3

• The behavior of a function can change from one side of a vertical asymptote to the other.