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 → a) f(x) = ∞ means, and why the limit still does not exist as a number. (Guide: “What an infinite limit means”; Practice Q5(c))
- I can explain why 1/x has no two-sided infinite limit at 0 while 1/x² does. (Guide: Figure 1; Practice Q3)
- I can state the limit-based definition of a vertical asymptote, including that one side is enough. (Guide: “Vertical asymptotes, defined with limits”; Practice Q2)
- I can explain why a vertical asymptote is a discontinuity that cannot be removed. (Guide: “Vertical asymptotes, defined with limits”; Practice Q6(d))
Skills
- I can use a sign check to find each one-sided limit when substitution gives a nonzero number over 0. (Guide: “Finding one-sided infinite limits: the sign check”; Practice Q1, Q4)
- I can predict whether the one-sided limits match from the power of the factor in the bottom. (Guide: Worked example 2; Practice Q3)
- I can factor and cancel to decide whether a zero of the bottom gives a hole or a vertical asymptote, and give the height of a hole. (Guide: Worked example 1; Practice Q2, Q6(a))
- I can find a parameter that turns a vertical asymptote into a hole, or that fixes the sign of an infinite limit. (Practice Q7)
- I can recall the infinite limits of ln x at 0 and tan x at π/2. (Guide: table of standard examples; Revision notes)
Reasoning
- I can write a justification sentence that names a one-sided infinite limit as the reason for a vertical asymptote. (Guide: Worked example 3; Practice Q6(c))
- I can read unbounded behaviour from a table and explain why a sign check is still needed to confirm it. (Practice Q5)
- I can explain why a point where f is undefined is not always a vertical asymptote. (Guide: Worked example 3, step 6)
All ticked? Move on to Topic 1.15, Connecting Limits at Infinity and Horizontal Asymptotes, or return to the Calculus AB roadmap or the Calculus BC roadmap.