limits - mrs. upham · 2020. 5. 28. · properties of limits some basic limits let b and c be real...
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LIMITS
Name: _________________________________________________________
Mrs. Upham
2019-2020
Lesson 1: Finding Limits Graphically and Numerically
When finding limits, you are finding the y-value for what the function is
approaching. This can be done in three ways:
1. Make a table
2. Draw a graph
3. Use algebra
Limits can fail to exist in three situations:
1. The left-limit is
different than
the right-side
limit.
𝑦 = |𝑥|
𝑥
2. Unbounded
Behavior
𝑦 = 1
𝑥2
3. Oscillating
Behavior
𝑦 = 𝑠𝑖𝑛 (1
𝑥)
Verbally: If f(x) becomes arbitrarily close to a single number L as x approaches c
from either side, then the limit of f(x) as x approaches c is L.
Graphically: Analytically:
Numerically: From the table, lim𝑥→−5
𝑓(𝑥) = 3.4
x -5.01 -5.001 -5 -4.999 -4.99
f(x) 3.396 3.399 3.4 3.398 3.395
1. Use the graph of f(x) to the right to find
lim𝑥→−3
2𝑥2 + 7𝑥+3
𝑥+3
2. Use the table below to find lim𝑥→2
𝑔(𝑥)
x 1.99 1.999 2 2.001 2.01
f(x) 6.99 6.998 ERROR 7.001 7.01
3. Using the graph of H(x), which statement is not true?
a. lim𝑥→𝑎−
𝐻(𝑥) = lim𝑥→𝑎+
𝐻(𝑥)
b. lim𝑥→𝑐
𝐻(𝑥) = 4
c. lim𝑥→𝑏
𝐻(𝑥) does not exist
d. lim𝑥→𝑐+
𝐻(𝑥) = 2
Lesson 2: Finding Limits Analytically
Properties of Limits
Some Basic Limits
Let b and c be real numbers and let n be a positive integer.
lim𝑥→𝑐
𝑓(𝑥) = 𝑓(𝑐)
lim𝑥→𝑐
𝑥 = 𝑐 lim𝑥→𝑐
𝑥𝑛 = 𝑐𝑛
Methods to Analyze Limits
1. Direct substitution
2. Factor, cancellation technique
3. The conjugate method, rationalize the numerator
4. Use special trig limits of lim𝑥→0
sin 𝑥
𝑥= 1 or lim
𝑥→0
1−cos 𝑥
𝑥= 0
Direct Substitution
1. lim𝑥→2
(3𝑥 − 5)
2. lim𝑥→4
√𝑥 + 43
3. lim𝑥→1
sin𝜋𝑥
2
4. lim𝑥→7
𝑥
5. If lim𝑥→𝑐
𝑓(𝑥) = 7 then lim𝑥→𝑐
5𝑓(𝑥)
6. lim𝑥→𝑐
√𝑓(𝑥)
7. lim𝑥→𝑐
[𝑓(𝑥)]2
8. Given: lim𝑥→𝑐
𝑓(𝑥) = 7 and lim𝑥→𝑐
𝑔(𝑥) = 4, find:
a. lim𝑥→𝑐
[𝑓(𝑥) + 𝑔(𝑥)]
b. lim𝑥→𝑐
𝑓(𝑔(𝑥))
c. lim𝑥→𝑐
𝑔(𝑓(𝑥))
Limits of Polynomial and Rational Functions:
9. lim𝑥→0
𝑥3+1
𝑥+1
10. lim𝑥→2
𝑥3+1
𝑥+1
11. lim𝑥→−1
𝑥3+1
𝑥+1
Limits of Functions Involving a Radical
12. lim𝑥→3
√𝑥+1−2
𝑥−3
Dividing out Technique
13. lim∆𝑥→
2(𝑥+ ∆𝑥)−2𝑥
∆𝑥
14. Given f(x) = 3x + 2
Find limℎ→0
𝑓(𝑥+ℎ)−𝑓(𝑥)
ℎ
15. lim𝑥→𝑐
sin 𝑥
16. lim𝑥→𝑐
cos 𝑥
17. lim𝑥→
𝜋
2
sin 𝑥
18. lim𝑥→𝜋
𝑥 cos 𝑥
19. lim𝑥→0
tan 𝑥
𝑥
20. lim𝑥→0
sin 3𝑥
𝑥
The Squeeze Theorem
If h(x) < f(x) < g(x) for all x in an open interval containing c, except possibly at c
itself, and if lim𝑥→𝑐
ℎ(𝑥) = 𝐿 = lim𝑥→𝑐
𝑔(𝑥) then lim𝑥→𝑐
𝑓(𝑥) exists and is equal to L.
4 – |𝑥| < f(x) < 4 + |𝑥|
Special Trigonometric Limits:
lim𝑥→0
sin 𝑥
𝑥= 1 lim
𝑥→0
1−cos 𝑥
𝑥= 0
Lesson 3: Continuity and One-Sided Limits
Definition of Continuity
Continuity at a point:
A function f is continuous at c if the following three conditions are met:
1. f(c) is defined
2. lim𝑥→𝑐
𝑓(𝑥) exists
3. lim𝑥→𝑐
𝑓(𝑥) = 𝑓(𝑐)
Properties of continuity:
Given functions f and g are continuous at x = c, then the following functions are
also continuous at x = c.
1. Scalar multiple: 𝑏 ° 𝑓
2. Sum or difference: f± g
3. Product: f • g
4. Quotient: 𝑓
𝑔 , if g(c) ≠ 0
5. Compositions: If g is continuous at c and f is continuous at g©, then the
composite function is continuous at c, (𝑓 ° 𝑔)(𝑥) = 𝑓(𝑔(𝑥))
The existence of a Limit:
The existence of f(x) as x approaches c is L if and only if lim𝑥→𝑐−
𝑓(𝑥) = 𝐿 and
lim𝑥→𝑐+
𝑓(𝑥) = 𝐿
Definition of Continuity on a Closed Interval:
A function f is continuous on the closed interval [a, b] if it is continuous on the
open interval (a, b) and lim𝑥→𝑎+
𝑓(𝑥) = 𝑓(𝑎) and lim𝑥→𝑏−
𝑓(𝑥) = 𝑓(𝑏)
Example 3: Given ℎ(𝑥) = {−2𝑥 − 5 ; 𝑥 < −2
3 ; 𝑥 = −2
𝑥3 − 6𝑥 + 3 ; 𝑥 > −2 for what values of x is h not
continuous? Justify.
Example 4: If the function f is continuous and if f(x) = 𝑥2−4
𝑥+2 when x ≠ -2, then
f(-2) = ?
Example 5: Which of the following functions are continuous for all real numbers x?
a. y = 𝑥2
3
b. y = ex
c. y = tan x
A) None B) I only C) II only D) I and III
Example 6: For what value(s) of the constant c is the function g continuous over all
the Reals? 𝑔(𝑥) = {𝑐𝑥 + 1 ; 𝑖𝑓 𝑥 ≤ 3
𝑐𝑥2 − 1 ; 𝑖𝑓 𝑥 > 3
Lesson 4: The Intermediate Value Theorem
The Intermediate Value Theorem (IVT) is an existence theorem which says that a
continuous function on an interval cannot skip values. The IVT states that if these
three conditions hold, then there is at least one number c in [a, b] so that f(c) = k.
1. f is continuous on the closed interval [a, b]
2. f(a) ≠ f(b)
3. k is any number between f(a) and f(b)
Example 1: Use the Intermediate Value Theorem to show that f(x) = 𝑥3 + 2x – 1 has
a zero in the interval [0, 1].
Example 2: Apply the IVT, if possible, on [0, 5] so that f(c) = 1 for the function
f(x) = 𝑥2 + 𝑥 − 1
Example 3: A car travels on a straight track. During the time interval 0 < t < 60
seconds, the car’s velocity v, measured in feet per second is a continuous function.
The table below shows selected values of the function.
t, in seconds 0 15 25 30 35 50 60
v(t) in ft/sec -20 -30 -20 -14 -10 0 10
A. For 0 < t < 60, must there be a time t when v(t) = -5?
B. Justify your answer.
Example 4: Find the value of c guaranteed by the Intermediate Value Theorem.
f(x) = x2 + 4x – 13 [0, 4] such that f(c) = 8
Lesson 5: Infinite Limits
Definition of Vertical Asymptotes:
A vertical line x = a is a vertical asymptote if lim𝑥→𝑎+
𝑓(𝑥) = ±∞ and/or
lim𝑥→𝑎−
𝑓(𝑥) = ±∞ ℎ(𝑥) = 𝑓(𝑥)
𝑔(𝑥) has a vertical asymptote at x = c.
Properties of Infinite Limits:
Let c and L be real numbers and let f and g be functions such that lim𝑥→𝑐
𝑓(𝑥) = ∞ and
lim𝑥→𝑐
𝑔(𝑥) = 𝐿
1. Sums or Difference: lim𝑥→𝑐
[𝑓(𝑥) ± 𝑔(𝑥)] = ∞
2. Product: lim𝑥→𝑐
[𝑓(𝑥)𝑔(𝑥)] = ∞ , 𝐿 > 0
lim𝑥→𝑐
[𝑓(𝑥)𝑔(𝑥)] = −∞ , 𝐿 < 0
3. Quotient: lim𝑥→𝑐
𝑐𝑔(𝑥)
𝑓(𝑥)= 0
Example 1: Evaluate by completing the table for lim𝑥→−3
1
𝑥2−9
x -3.5 -3.1 -3.01 -3.001 -3 -2.999 -2.99 -2.9 -2.5
f(x)
Example 2: Evaluate lim𝑥→1
1
(𝑥−1)2
Example 3: Evaluate lim𝑥→1+
𝑥+1
𝑥−1
Example 4: Evaluate lim𝑥→1+
𝑥2−3𝑥
𝑥−1
Example 5: Evaluate lim𝑥→1+
𝑥2
(𝑥−1)2
Example 6: Evaluate lim𝑥→0−
(𝑥2 − 1
𝑥)
Example 7: Evaluate lim𝑥→(
−1
2)
+
6𝑥2+𝑥−1
4𝑥2−4𝑥−3
Example 8: Find any vertical asymptotes or removable discontinuities 𝑓(𝑥) = 𝑥−2
𝑥2−𝑥−2
Example 9: Determine whether the graph of the function has a vertical asymptote
or a removable discontinuity at x = 1. Graph the function to confirm
𝑓(𝑥) = sin(𝑥 + 1)
𝑥 + 1