AP Calculus AB Infinite Limits and Asymptotes Test with Point Values (PDF)

Find \(\lim_{x\to 5^+} \frac{-2}{x-5}\) and \(\lim_{x\to 5^-} \frac{-2}{x-5}\).

Find all vertical asymptotes of \(f(x)=\frac{x-1}{x^{2}-1}\) and state where there is a hole.

Find each limit. (a) \(\lim_{x\to\infty}\frac{7x^{2}-x}{2x^{2}+5}\) (b) \(\lim_{x\to\infty}\frac{3x-4}{x^{3}+1}\) (c) \(\lim_{x\to\infty}\frac{x^{4}}{x^{2}+9}\).

Find \(\lim_{x\to\infty}\frac{2x}{\sqrt{x^{2}+1}}\) and \(\lim_{x\to-\infty}\frac{2x}{\sqrt{x^{2}+1}}\).

The graph shows \(g(x)=\frac{1}{(x+2)^{2}}+1\). (a) State \(\lim_{x\to -2} g(x)\). (1 point) (b) State \(\lim_{x\to\infty} g(x)\) and \(\lim_{x\to-\infty} g(x)\). (1 point) (c) Write the equations of both asymptotes. (1 point)

Let \(r(x)=\frac{3x+3}{x-2}\). Find the vertical asymptote, the horizontal asymptote and both intercepts. Use the graph to check.

A bakery’s average cost per loaf is \(A(n)=\frac{3n+90}{n}\) dollars when it bakes \(n\) loaves. (a) Find \(A(30)\). (1 point) (b) Find \(\lim_{n\to\infty}A(n)\) and interpret it. (1 point)

The speed of a cooling fan in rpm is modeled by \(D(t)=\frac{80t}{t+4}\) after \(t\) seconds. (a) Find \(D(4)\). (1 point) (b) Find \(\lim_{t\to\infty}D(t)\). (1 point) (c) Explain what the limit means for the fan. (1 point)