Work through the list without notes or a calculator. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. This topic is shared by Calculus AB and Calculus BC.
Understanding
- I can explain what lim (x → ∞) f(x) = L means, using a table of large x values. (Guide: “What x → ∞ means”)
- I can describe the three kinds of end behaviour: levelling off, growing without bound, and oscillating. (Guide: “End behaviour and horizontal asymptotes”; Practice Q4)
- I can say when a line y = L is a horizontal asymptote, and why a graph has at most two. (Guide: “End behaviour and horizontal asymptotes”; Practice Q4, Q7(c))
- I can explain why a graph may cross its horizontal asymptote, with an example. (Guide: Figure 1; Practice Q5(d))
Skills
- I can find a limit at infinity of a rational function by dividing by the highest power of x in the denominator. (Guide: Worked example 1; Practice Q1, Q5(a))
- I can predict the limit of a rational function from the degrees and leading coefficients. (Guide: degree table; Practice Q1, Q3(D))
- I can handle square roots as x → −∞ using √(x²) = |x|. (Guide: Worked example 2; Practice Q2)
- I can find where a graph meets its horizontal asymptote by solving f(x) = L. (Guide: Worked example 1; Practice Q5(c), Q7(d))
- I can use the growth order ln x < xᵖ < bˣ to evaluate a limit at infinity. (Guide: Worked example 3; Practice Q3, Q7(a))
Reasoning
- I can compare two growing functions by finding the limit of their ratio, and state which dominates. (Guide: “Comparing how fast functions grow”; Practice Q3)
- I can interpret a limit at infinity in a context, with units. (Practice Q6(a), Q6(b))
- I can tell a horizontal asymptote (x → ±∞) apart from a vertical asymptote (f(x) → ±∞). (Guide: Worked example 1, “Connection to Topic 1.14”)
All ticked? Move on to Topic 1.16, Working with the Intermediate Value Theorem, or return to the Calculus AB roadmap or the Calculus BC roadmap.