Limits on Paper: Tables, Patterns, Algebra

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Mathematics · Precalculus / Trigonometry
SP-PRECAL-005 · Free to print and use in your classroom · substituteplans.com
Page 2
Limits on Paper: Tables, Patterns, Algebra
Mathematics · Precalculus / Trigonometry · Grades 11-12 · 50 minutes
Time needed
50 minutes (one class period)
Prep
Copy the 2-page worksheet (one per student).
Technology
None. Calculators are optional.
Student
independence
High. Students work alone, then compare with a partner.
Copies to make
One 2-page worksheet per student
Materials
Pencils, Worksheet
Goal for today: Students will estimate limits from tables, decide when a limit does not exist, and evaluate simple
limits using substitution and factoring.
Before the bell
☐ Copy the worksheet, one per student (2 pages).
☐ Write on the board: A limit asks, “What value does f(x) approach as x approaches a?”
☐ Keep the answer key on your desk. You do not need to solve anything yourself.
Say this to start
Good morning! Today is an introduction to limits, the key idea behind calculus. A limit asks what value a function gets
close to as the input gets close to a number. You will use tables of values and a little algebra, and everything you need
is in the Key Idea box. Work on your own first, and then we will check answers.
Run of show
Minutes
What you do
What students do
0–5
Take attendance. Hand out the worksheet. Write the board
question about approaching a value.
Settle in and read the Key Idea box.
5–10
Read the first example aloud: f(1.9) = (3.61 − 4)/(−0.1) =
3.9. Show how the numerator and denominator are
computed.
Follow along and check the example.
10–22
Students complete the Part A table and questions 1–2.
Calculators are fine for arithmetic.
Compute three values and describe the
pattern.
22–30
Students do Part B (questions 3–5). Remind them: left side
uses x less than 2; right side uses x greater than 2.
Read one-sided limits from the tables.
30–42
Students do Part C (questions 6–9) and Part D (questions
10–12).
Find limits by substitution and factoring,
then choose answers.
42–50
Read answers from the key. Students mark their work.
Collect worksheets.
Check answers and hand in the worksheet.

SubstitutePlans.com
Mathematics · Precalculus / Trigonometry
SP-PRECAL-005 · Free to print and use in your classroom · substituteplans.com
Page 3
If things go wrong
• If students say they have not learned limits, point them to the Key Idea box. The tables show the pattern
without any calculus.
• If a student says f(2) must be 4, remind them the table lists f(2) as undefined; the limit tells us what f(x) is
heading toward, not what happens at 2.
• If students get 0/0 in Part C, tell them to factor the top and cancel the matching factor.
• If a student finishes early, use the early finisher list.
Early finishers
• Make your own table of values (4 inputs) for f(x) = (x² − 1)/(x − 1) near x = 1 and find the limit.
• Write one sentence explaining why the limit can exist when f(a) does not.
Support and challenge
• Support: Students complete Parts A and B and questions 6, 7 and 10.
• Challenge: Ask students to find lim(x→−3) (x² + 5x + 6)/(x + 3) and explain each step.
Notes for the teacher (you fill this in)
Jot a few lines before you leave. A full Daily Sub Report is in the free Sub Survival Kit at SubstitutePlans.com.
How far the class got / what to pick up next:
Students who were especially helpful:
Anything the teacher should know: