Functions with Asymptotes and Holes (8.5.4)

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

• A hole (or point discontinuity) occurs in the graph of a function f at a point c if lim x->c f(x) exists and f(c) is undefined or not equal to lim x->c f(x).

Answer 3

• A hole (or point discontinuity) occurs in the graph of a function f at a point c if lim x->c f(x) exists and f(c) is undefined or not equal to lim x->c f(x).