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 sum, difference, constant multiple, product, quotient, power and root properties, with the condition each one needs. (Guide: “The properties of limits”; Practice Q1)
- I can explain why the quotient property cannot be used when the bottom limit is 0, and what nonzero/0 tells me. (Guide: Worked example 1(d); Practice Q4)
- I can explain why substitution gives the limit of a polynomial, using the properties step by step. (Guide: “Why direct substitution works for polynomials”)
- I can explain why a limit can differ from the function value at that point. (Guide: Worked example 3; Practice Q5(b))
Skills
- I can combine given limit values to find the limit of a more complicated expression, naming each property. (Guide: Worked example 1; Practice Q1, Q7(b))
- I can find the limit of a polynomial or rational function by substitution, after checking the bottom is not 0. (Practice Q2)
- I can find the limit of a composite function, and track the side of approach when the outer function jumps. (Guide: “Limits of composite functions”; Practice Q3, Q5(d))
- I can find one-sided limits from a piecewise formula and choose a constant that makes the two-sided limit exist. (Guide: Worked example 3; Practice Q6)
- I can read one-sided limits of sums and products from two graphs, treating open and filled circles correctly. (Guide: Worked example 2; Practice Q5)
Reasoning
- I can show that a sum can have a limit even when neither part has one. (Guide: Worked example 2(c); Practice Q7(d))
- I can use the difference property to find an unknown limit, or to prove that a limit cannot exist. (Practice Q7(a), Q7(c))
All ticked? Move on to Topic 1.6, Determining Limits Using Algebraic Manipulation, or return to the Calculus AB roadmap or the Calculus BC roadmap.