Lesson 14: Finding Limits Algebraically
Lesson Objectives:
A calculus student should know how to find the limit of a given equation by using
algebraic steps such as substitution and factoring.
At the end of the lesson the student will know about
1. Finding limits algebraically
2. Finding limits involving infinity or nonexistent
Pre-Requisite Knowledge:
There are three steps to finding limits algebraically:
1. Substitute x with the given limit
2. If step 1 doesn’t work, then factor and cancel common expressions
3. If the equation involves square roots, then you should rationalize
These are special trig limits that students should learn:
1. lim (sin x) /x =1
x 0
2. lim (1-cos x) /x =0
x 0
Warm-up:
Since this lesson will involve steps such as substitution, here is a question to get students
familiarized with similar steps.
If y=3, find the value of x in the equation 9x2-4xy+6=0.
Lesson Plan:
 Watch the brighstorm videos (part 1&2) with the students so that they get a
glimpse of what the lesson will be about. (5-6 mins.)
 Go over the seymour worksheet with the students. (10-15 mins.)
 Along with students, do the MMM worksheet Unit 4 pgs.28-29 examples 1-11
(15-20 mins.)
 Finally go over the trig limits power point.
Homework:
Textbook: Read pgs.108-114 and do #1, 3, 6, 13, and 31
MMM Worksheet: Pgs.30-31 #1-27
Enrichment Plan:
 Students may watch a video Introduction to limits of equations by Ms. Roshan to
help them under stand the homework.
 They can also go over the limit laws which are attached at the end of the lesson.
 Students can use this website to review the tutorials to solving limits.
 At the end of the lesson students should be able to complete the teacherweb
worksheet.
 Finally, students are recommended to take the limits quiz also attached at the end.
Limit Laws:
If the limits
and
both exist, then:
1. Addition Law:
2. Subtraction Law:
3. Constant Law:
4. Multiplication Law:
5. Division Law:
6. Power Law: If n is an integer, and the limit
7. Root law: If n is an integer, the limit
even, then
8. Squeeze Law: If
exists, then
exists, and that limit is positive if n is
for all x in an open interval that contains a,
except possibly at a itself, and
9. Composition Law: If f is continuous at b and
, then
, then
Limits Quiz
1. Evaluate the limit if it exists:
a. 5
b. 3
c. -3
d. -5
2. Evaluate the limit if it exists:
a. 1/4 b. -1/4 c. 1
d. 0
3. Evaluate the limit if it exists:
a. 1/4 b. -1/4 c. 1
d. -1
4. Evaluate the limit if it exists:
a. 0
lim x2+x-6
x 2
2x
e. DNE
lim √(x-5)-2
x 9
x-9
e. DNE
lim 1/x-1/2
x 2
x-2
e. DNE
lim tan-1x .
x 2
sin-1x+1
b. 1/4 c. 1/2 d. π/2 e. π/2π+4