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 state the squeeze theorem, including where the inequality must hold. (Guide: “The squeeze theorem”; Practice Q1)
- I can explain why the product law cannot find lim (x → 0) x² sin(1/x). (Guide: “When rewriting is not enough”; Practice Q6(a), Q6(d))
- I can explain why the squeeze theorem gives the limit at a but not the value f(a). (Guide: Worked example 2; Practice Q5(b))
- I can say what follows when the two outer limits are different: no conclusion. (Guide: Worked example 2(c); Practice Q3, Q5(c))
Skills
- I can build bounds from −1 ≤ sin u ≤ 1 by multiplying by x² or |x| and adding a constant. (Guide: “How to build the bounds” and Worked example 1)
- I can use absolute values when a factor such as x or x − 3 can be negative. (Guide: “How to build the bounds”; Practice Q3, Q6(b))
- I can find a limit with the squeeze theorem and name all three conditions. (Guide: Worked example 1; Practice Q5(a), Q6(c))
- I can use lim (x → 0) (sin x)/x = 1 and lim (x → 0) (1 − cos x)/x = 0 to find related limits, matching the angle and the denominator. (Guide: Worked example 3; Practice Q2, Q4, Q7(c))
Reasoning
- I can explain, with the unit-circle areas, why cos x ≤ (sin x)/x ≤ 1 near 0 and why that gives the limit 1. (Guide: “Result 1” and Figure 2)
- I can turn a given inequality into a new one, such as the bounds for (tan x)/x, and keep the signs right. (Practice Q7(a), Q7(b))
- I can check a squeeze answer with one nearby value and explain why the check is not a proof. (Guide: Worked examples 1 and 3; Practice Q7(d))
- I can explain why the trig limits need radians. (Guide: “Degrees warning”)
All ticked? Move on to Topic 1.9, Connecting Multiple Representations of Limits, or return to the Calculus AB roadmap or the Calculus BC roadmap.