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 why 0/0 is an indeterminate form and give two 0/0 expressions with different limits. (Guide: “Why direct substitution sometimes fails”; Practice Q1)
- I can explain why cancelling a factor (x − a) does not change the limit as x → a. (Guide: “The key fact”; Practice Q6(b))
- I can explain the difference between lim (x → a) f(x) and f(a), using a graph with a hole. (Guide: Figure 1; Practice Q6(c), Q6(d))
- I can tell 0/0 apart from a nonzero number over 0, and say what each means. (Guide; Practice Q5)
Skills
- I can factor a quadratic, a difference of squares or a difference of cubes to cancel (x − a). (Guide: Worked example 1; Practice Q1, Q6)
- I can multiply by a conjugate, expanding only the side with the root, to find a limit. (Guide: Worked example 2; Practice Q2, Q7)
- I can combine fractions inside a fraction and cancel to find a limit, handling a − x = −(x − a). (Guide: “Technique 3 in action”; Practice Q3)
- I can use sin²x = (1 − cos x)(1 + cos x) or sin 2x = 2 sin x cos x to find a trig limit. (Guide: “Technique 4 in action”; Practice Q4)
- I can keep “lim” on every line until the substitution step. (Guide: “A note on notation”)
Reasoning
- I can state the condition that makes each technique appropriate before I use it. (Guide: technique table; Revision notes)
- I can check an exact limit with one nearby value, and explain why the check is not a proof. (Guide: Worked examples 1 and 2; Practice Q7(c))
- I can explain why the limit as x → 0 of (sin x)/x cannot be found by cancelling, and name the topic that handles it. (Guide: “Pointer to Topic 1.8”)
All ticked? Move on to Topic 1.7, Selecting Procedures for Determining Limits, or return to the Calculus AB roadmap or the Calculus BC roadmap.