Unit 1 Notes:
Limits and Continuity
Learning Target 1-1: The Difference Quotient.
Objective: Use the difference quotient to find slopes and equations of
tangent lines.
1-1
Warm Up:
A. Write the equation of the line passing through the points ( −3 , 4 ) and ( 2 , −8 ) in slope
intercept form.
B. Write the equation of the line that is perpendicular to the line in Part A and passes through
the point (
1
4
2
, − ) in slope intercept form.
3
Concept:
1
Graph the equation 𝑓(𝑥) = − (𝑥 − 4)2 + 4 by hand.
4
Average Rate of Change =
Instantaneous Rate of Change =
Ex. 1: Find the difference quotient for the given equation, then find the slope of the line
tangent to the curve at 𝑥 = 2 . Write the equation of the tangent line at that 𝑥 value.
ℎ(𝑥) = 𝑥 2
Ex. 2: Write the equation, in standard form, of the line tangent to 𝑔(𝑥) =
1
𝑥
at 𝑥 = −2 .
Ex. 3: Find the difference quotient for the concept function. Check your work by substituting at
least two different values for 𝑥 and verifying the slope is correct utilizing the graph.
1
𝑓(𝑥) = − (𝑥 − 4)2 + 4
4
Learning Target 1-2: Evaluating Limits from Tables and Graphs.
Objective: Evaluate limits from tables and graphs, including one-sided limits and
limits at infinity.
1-2
The equation of the function graphed to the right is
f (x) =
2x2 + 5x − 3
x2 - 9
. The coordinates of the hole in
7
the graph are (-3, 6)
Statements
Calculus Limit Notation
As x → −∞, the graph of f(x) → _________.
As x → ∞, the graph of f(x) → _________.
As x → –3 from the left, the graph of f(x) → _________.
As x → –3 from the right, the graph of f(x) → _________.
As x → –3 from both sides, the graph of f(x) → _________.
As x → 3 from the left, the graph of f(x) → _________.
As x → 3 from the right, the graph of f(x) → _________.
As x → 3 from both sides, the graph of f(x) → _________.
Based on what you have just seen, how might you informally define what the value of a limit represents in terms of the
graph?
Limits That May Not Exist
Example #1
Find each of the following from the graph.
a) lim f (x) =
x®2
b) lim f (x) =
+
x®2
c) f(2) =
d) Does lim f (x) exist or not? Why or why not?
x®2
Example #2
Find each of the following from the graph.
a) lim f (x) =
x®2
b) lim f (x) =
+
x®2
c) f(2) =
d) Does lim f (x) exist or not? Why or why not?
x®2
Example #3
Find each of the following from the graph.
a) lim f (x) =
x®2
b) lim f (x) =
+
x®2
c) f(2) =
d) Does lim f (x) exist or not? Why or why not?
x®2
Based on what you have seen so far, does f(a) have to be defined in order for the lim f (x) to exist? Draw and explain
x® a
two different graphs to justify your reasoning. In both graphs, f(a) should be undefined but in one graph, the limit should
exist while in the second one, it should not exist.
First Graph
Second Graph
Finding Limits From Tables
Example #1: Below is a table of values of an exponential function. Use the table to find the limits that follow.
x
f(x)
a) lim f (x) =
-9
513
-5
33
-3
9
b) lim f (x) =
x®¥
x®-3
-1
3
1
1.5
3
1.125
9
1.002
c) lim f (x) =
d) lim f (x) =
x®1
x®-¥
Example #2: Below is a table of values of a rational function. Use the table to find the limits that follow.
x
F(x)
-1000
1.002
-1.001
2001
-1
Undefined
-0.999
-1999
b)
d) lim f (x) =
e) lim f (x) =
x®2
1.999
0.333
lim f (x) =
a) lim f (x) =
x®¥
0
-1
c) lim f (x) =
+
x®-1-
x®-¥
2
Undefined
x®-1
f) lim f (x) =
x®-1
2.001
0.334
1000
0.998
Practice:
2)
1)
a) lim f (x)
-
b) lim f (x)
+
a) lim f ( x )
b) lim f ( x )
c) lim f (x)
d) f (0)
c) lim f ( x )
d) f (2)
e) lim f ( x)
f) lim f ( x)
e) lim f ( x)
f) lim f ( x)
x®0
x®0
x  
x®0
x 
x2
x2
x  
x2
x 
4)
3)
a) lim f (x)
-
b) lim f (x)
+
a) lim f (x)
-
b) lim f (x)
+
c) lim f (x)
d) f (2)
c) lim f (x)
d) f (0)
e) lim f ( x)
f) lim f ( x)
e) lim f ( x)
f) lim f ( x)
x®2
x®2
x  
x®2
x 
x®0
x®0
x  
x®0
x 
5)
x
f(x)
1.9
2.99
1.99 1.999
2.999 2.9999
a) lim f ( x )
2
2.001
4.999
2.01
4.99
b) lim f ( x )
x2
2.1
4.75
c) lim f ( x )
x2
x2
d) f (2)
6)
x
f(x)
5.9
56.5
a) lim f (x)
x®6
5.99 5.999
56.75 56.99
6
6.001
56.999
b) lim f (x)
+
x®6
6.01
56.85
6.1
56.5
c) lim f (x)
x®6
d) f (6)
Learning Target 1-3: Algebraic Limits.
1-3
Objective: Evaluate finite limits using algebraic analysis.
Limit Laws
1. lim  f ( x)  g ( x)
x a
2. lim  f ( x)  g ( x)
x a
f ( x)
x a g ( x)
3. lim
4. lim  f ( x)  g ( x)
x a
5. lim c  f ( x)
x a
6. lim  f ( x) p
x a
7. lim c
x a
 f ( g ( x )) 
8. lim
x a
Graph of f(x)
Graph of g(x)
Find each of the following limits applying the properties of limits. If a limit does not exist, state
why.
lim  f ( x)  g ( x)
x2
 2 f ( x)
x  6 g ( x)
lim
lim
x2

2 g ( x)
lim 2 f ( x)  3g ( x)
x  1
lim 2 f ( x) g ( x)
lim  f ( x)  g ( x)
x  3
lim  f ( x)2
x  2
x 4
lim
x 2
f ( x)
g ( x)
lim  f ( g ( x )) 
x  5
1
Consider the function f(x)= 2 |−2𝑥 + 4| − 4, for a moment. The graph of f(x) is pictured below. From the graph,
determine the following limits.
Find f(a) using the
equation.
lim f (x)
x®a
Find lim f (x)
x®a
from the graph.
lim f (x)
x®0
lim f (x)
x®2+
lim f (x)
x®10
When a function is defined and continuous at a value, x = a, how can lim f (x) be found analytically?
x®a
The easiest way to find a limit algebraically is to PLUG IT IN i.e SUBSTITUTION.
Ex.
f(x) = x2 + 1
a) lim f (x)
x®0
b) lim f (x)
x®2
c) lim f (x)
x®5
However, when direct substitution gives you an undefined value there are three other methods that are great options
to try.
Method #1: Factor
Ex. 1) lim
x®3
x2 - 9
x-3
Method #2: Find a common denominator
Ex.
1
-1
x
4
lim
x®5
x -5
5x 2 - 5x
x®1
x -1
Ex. 2) lim
x+2
x®-2 x 2 - x - 6
Ex. 3) lim
Method #3: Multiply by the conjugate
Ex.
x +1 - 2
x -3
lim
x®3
Here is a picture of what this graph looks like:
Practice: Find each of the following limits analytically. Show your algebraic analysis.
ln x
x®e 2x
a)
lim
b)
æ2
ö
lim- ç x 2 + 2x ÷
x®5 è 5
ø
c)
(
lim sin 2 q + 2cosq
q ®p
lim
5p
3
tan a
a2
d)
a®
e)
x2 - x - 6
x®-2 2x + 4
lim
)
lim
f)
g)
h)
i)
j)
k)
x®3
x +5
x2 - 9
8x 3 - 27
lim
3 2x - 3
x®
2
lim
x®-2
lim
x®1
2x + 5 -1
x+2
2 - 3+ x
x -1
1
-1
lim x - 5
x®6
x-6
lim+
x®2
3x 2 + 7x + 2
x2 - 4
Learning Target 1-4: Continuity.
Objective: Determine if a function is continuous. If it is not, state the
type(s) of discontinuity.
1-4
For the function graphed below, fill in the table on the bottom of this page by answering the questions.
After filling in the table, write three pieces of information (in pencil) that must be true in order for a
function, G(x), to be continuous at x = a.
1.
2.
3.
x=a
x = -6
x = -3
x=0
x=2
x=6
Is the function
defined? If so,
what is its
value?
What is the
value of
What is the
value of
What is the
value of
limG(x)?
limG(x)?
limG(x)?
x->a-
x->a+
x->a
Is G(x)
continuous
at x = a?
x=8
The graph of the function, G(x), pictured to the right has several x – values at which the function is not
continuous. For each of the following x – values, use the three part definition of continuity to determine if
the function is continuous or not.
1.
x = -8
2.
x = -6
3.
x = -4
#4-#5 Directions: Use the three part definition of continuity to determine if the given functions are
continuous at the indicated values of x.
𝑒 𝑥 𝑐𝑜𝑠𝑥,
4.
−2√𝑥 + 6,
𝑥 < −2
𝑓(𝑥) = {3𝑥 + 2,
𝑥 = −2 at x = -2
𝑒 𝑥 + cos(𝜋𝑥) , 𝑥 > −2
6.
Consider the function, f(x), to the right to answer the following questions.
5.
𝑔(𝑥) = {
3𝜋
𝑒 𝑥 𝑡𝑎𝑛 ( 4 ) ,
𝑥<𝜋
𝑥≥𝜋
at x = 𝜋
a. What two limits must equal in order for f(x) to be continuous at x = -1?
x ≤ -1
2,

f ( x)  mx  k , -1 < x < 3
 2,
x≥3

b. What two limits must equal in order for f(x) to be continuous at x = 3?
c. Determine the values of m and k so that the function is continuous everywhere.
7. Sketch a possible graph for a function f that has the stated properties. f(-2) exists, the limit as x
approaches -2 exists, f is not continuous at x = -2, and the limit as x approaches 1 does not exist.
8.
Consider the function, g(x), to the right to answer the following questions.
a. What two limits must equal in order for g(x) to be continuous at x = -2?
kx2  m, x < -2

g ( x)  4 x  1, -2 ≤ x ≤ 3
kx  m,
x> 3

b. What two limits must equal in order for g(x) to be continuous at x = 3?
c. Determine the values of m and k so that the function is continuous everywhere.
9. Draw a sketch of a graph that meets the following requirements. The limit does not exist as x approaches -2.
The functional value at x = 3 is equal to the limit of f as x approaches 3. The function increases without bound
as x decreases without bound.
Learning Target 1-5: Intermediate Value Theorem (IVT).
Objective: Use the intermediate value theorem (IVT).
1-5
As we study calculus, we will study several different theorems. The first theorem of investigation
is the Intermediate Value Theorem.
Intermediate Value Theorem
Now, investigate the graphs below to determine if the theorem is applicable for these functions on the
specified intervals for the values given.
𝑓(𝑥) = {


−(𝑥 + 3)2 + 4, 𝑥 < −2
−(𝑥 + 3)2 + 4, 𝑥 < −2
𝑓(𝑥)
=
{

1
1
− 2 𝑥 − 1,
𝑥 > −2
− 2 𝑥 − 1,
𝑥 > −2


Is there a value of c on [-3, 0] such that f(c) = 2 ?
Is there a value of c on [-1, 5] such that f(c) = 2 ?
Does the I.V.T. guarantee a value of c such that
f(c) = 2 on the interval [–3, 0]? Why or why not?
Does the I.V.T. guarantee a value of c such that
f(c) = 2 on the interval [–1, 5]? Why or why not?
What two conditions must be true to verify the applicability of the Intermediate Value Theorem?
1.
2.
For each of the following functions, determine if the I.V.T. is applicable or not and state why or why not.
Then, if it is applicable, find the value of c guaranteed to exist by the theorem.
𝑥−3
2
1.
𝑓(𝑥) = 𝑥+2 on the interval [-1, 3] for f(c) = 3
3.
𝑓(𝑥) = 𝑥−2 on the interval [-1, 1] for f(c) = − 2
𝑥
𝑥−3
2
2.
𝑓(𝑥) = 𝑥+2 on the interval [-4, 1] for f(c) = 3
4.
𝑓(𝑥) = − (2)
for f(c) = -4
1
1 −𝑥+3
− 2 on the interval [3, 5]
5. Use the function h(x) below to answer the following questions. For each of the following points, decide
whether the function is or is not continuous and explain why. Use proper notation in your explanations.
a. ) x = -4
b.) x = -2
c.) x = 1
d.) x = 3
e.) x = 5
Learning Target 1-6: End Behavior Models (EBM).
Objective: Use an end behavior model (EBM), evaluate infinite limits, identify
horizontal asymptotes.
1-6
Determine a rational monomial end behavior model for each of the following equations,
evaluate the limits given without a calculator, and state any horizontal asymptotes.
12𝑥 3 −7
1. 𝑓(𝑥) = 12𝑥+11
2. 𝑓(𝑥) =
EBM:
EBM:
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
𝑥→−∞
HA:
5𝑥 2 −4
HA:
4. 𝑓(𝑥) = 2𝑥 3 −𝑥 2 +𝑥
5. 𝑓(𝑥) =
EBM:
EBM:
lim 𝑓(𝑥) =
𝑥→∞
lim 𝑓(𝑥) =
𝑥→−∞
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
8. f (x) =
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
𝑥→−∞
lim 𝑓(𝑥) =
HA:
3x 3 + 30
2x 3
9. f (x) =
EBM:
lim 𝑓(𝑥) =
𝑥→∞
lim 𝑓(𝑥) =
𝑥→−∞
HA:
5𝑥
𝑥→∞
𝑥→−∞
1
x-5
cos 𝑥−𝑥 3
EBM:
HA:
𝑥→∞
HA:
6. 𝑓(𝑥) =
𝑥
lim 𝑓(𝑥) =
EBM:
𝑥→−∞
3𝑥+sin 𝑥
𝑥→∞
HA:
6𝑥 2 −6
𝑥→∞
𝑥→−∞
HA:
−4𝑥+5+6𝑥2
EBM:
𝑥→∞
𝑥→−∞
3. 𝑓(𝑥) =
−1+8𝑥
lim 𝑓(𝑥) =
𝑥→∞
7. f (x) =
3𝑥 2 −5𝑥+1
EBM:
lim 𝑓(𝑥) =
𝑥→∞
lim 𝑓(𝑥) =
𝑥→−∞
HA:
7x
9x + 2x 2 + 4
10. f (x) =
2
-8x + x 2
11. f (x) =
EBM:
6x 2 + 3
x -1
12. f (x) =
EBM:
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
𝑥→∞
𝑥→∞
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
𝑥→−∞
𝑥→−∞
HA:
EBM:
lim 𝑓(𝑥) =
𝑥→∞
lim 𝑓(𝑥) =
𝑥→−∞
HA:
13. f (x) =
3x 2 -1
x3
14. f (x) =
EBM:
HA:
12x 4 +10x - 3
3x 4
15. f (x) =
EBM:
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
𝑥→∞
𝑥→∞
lim 𝑓(𝑥) =
lim 𝑓(𝑥) =
𝑥→−∞
𝑥→−∞
HA:
5x 3 - 4
4x - 5 + x 2
13x 4 + x 2
6x + 3
EBM:
lim 𝑓(𝑥) =
𝑥→∞
lim 𝑓(𝑥) =
𝑥→−∞
HA:
HA:
16. Sketch the graph of a function 𝑓(𝑥) that satisfies all of the following conditions:
y
a. lim 𝑓(𝑥) = −1
b.
c.
𝑥→2
𝑥→4
lim+ 𝑓(𝑥) = −∞
𝑥→4
𝑥→∞
lim
𝑥→−∞
7
6
lim− 𝑓(𝑥) = ∞
d. lim 𝑓(𝑥) = ∞
e.
8
𝑓(𝑥) = 2
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1–1
1
2
3
4
5
6
8 x
7
–2
–3
–4
–5
–6
–7
–8
17. Sketch the graph of a function 𝑓(𝑥) that satisfies all of the following conditions:
y
a. lim 𝑓(𝑥) = 2
b.
c.
d.
e.
𝑥→1
lim− 𝑓(𝑥) = ∞
𝑥→5
lim 𝑓(𝑥) = ∞
𝑥→5+
lim + 𝑓(𝑥) = −∞
𝑥→−2
lim − 𝑓(𝑥) = ∞
𝑥→−2
f. lim 𝑓(𝑥) = −1
𝑥→∞
8
7
6
5
4
3
2
1
–8 –7 –6 –5 –4 –3 –2 –1–1
–2
–3
–4
–5
–6
–7
–8
1
2
3
4
5
6
7
8 x
g.
lim
𝑥→−∞
𝑓(𝑥) = 0