Chapter 1
Limits and Their Properties
Section 1.2: Finding Limits Graphically & Numerically
Calculus: Chapter 1 – Section 1.2
Section 1.2: Finding Limits Graphically & Numerically
Calculus: Chapter 1 – Section 1.2
Section 1.2: Finding Limits Graphically & Numerically
Limits that Fail to Exist
Calculus: Chapter 1 – Section 1.2
Section 1.2: Finding Limits Graphically & Numerically
Limits typically do not exists when:
•f(x) approaches a different number from the right
side of c than it approaches from the left side
•f(x) increases or decreases without bound as x
approaches c
•f(x) oscillates between two fixed values as x
approaches c
Calculus: Chapter 1 – Section 1.2
Section 1.2: Finding Limits Graphically & Numerically
- Definition of a Limit
Let f be a function on an open interval containing
c (except possibly at c) and let L be a real
number. The statement
lim f ( x)  L
x c
Means that for each  > 0, there exists a  > 0
such that if
0  xc   
Calculus: Chapter 1 – Section 1.2
f ( x)  L  
Section 1.2: Finding Limits Graphically & Numerically
- Definition of a Limit
Calculus: Chapter 1 – Section 1.2
Section 1.3: Evaluation Limits Analytically
SECTION 1.3
EVALUATING LIMITS ANALYTICALLY
Calculus: Chapter 1 – Section 1.3
Section 1.3: Evaluation Limits Analytically
Properties of Limits
One of the easiest and most useful ways to
evaluate a limit analytically is direct substitution
If you can plug c into f(x) and generate a real
number answer in the range of f(x), that generally
implies that the limit exists
Basic Limits:
lim b  b
x c
Calculus: Chapter 1 – Section 1.3
lim x  c
x c
lim x  c
n
x c
n
Section 1.3: Evaluation Limits Analytically
Properties of Limits
Let b and c be real numbers, let n be a positive
integer, and let f and g be functions with the
following limits:
lim f ( x)  L
x c
Calculus: Chapter 1 – Section 1.3
lim g ( x)  K
x c
Section 1.3: Evaluation Limits Analytically
Properties of Limits
1. Scalar Multiple:
lim b  f ( x)  b  L
2. Sum/Difference:
lim  f ( x)  g ( x)  L  K
3. Product:
lim  f ( x)  g ( x)   L  K
4. Quotient:
f ( x)
L
lim

,
x c g ( x)
K
5. Power:
Calculus: Chapter 1 – Section 1.3
x c
x c
x c
lim  f ( x)   Ln
n
x c
K 0
Section 1.3: Evaluation Limits Analytically
Limits of Polynomials & Rationals
If p is a polynomial function and c is a real
number, then:
lim p( x)  p(c)
x c
If r is a rational function given by r(x) = p(x)/q(x), and
c is a real number, then
p (c )
lim r ( x)  r (c) 
, q (c )  0
x c
q (c )
Calculus: Chapter 1 – Section 1.3
Section 1.3: Evaluation Limits Analytically
Limits of Radical Functions
Let n be a positive integer. The following limit is
valid for c > 0 if n is even:
lim x  c
n
x c
Calculus: Chapter 1 – Section 1.3
n
Section 1.3: Evaluation Limits Analytically
Limits of Composite Functions
If f and g are functions such that
lim g ( x)  L and
x c
lim f ( x)  f ( L)
xL
then,


lim f ( g ( x))  f lim g ( x)  f ( L)
x c
Calculus: Chapter 1 – Section 1.3
x c
Section 1.3: Evaluation Limits Analytically
Limits of Trig Functions
Let c be a real number in the domain of the given
trig function:
1. limsin c  sin c
2. lim cos c  cos c
3. lim tan c  tan c
4. lim cot c  cot c
5. limsec c  sec c
6. lim csc c  csc c
x c
x c
x c
Calculus: Chapter 1 – Section 1.3
x c
x c
x c
Section 1.3: Evaluation Limits Analytically
Stratagies for Finding Limits
To find limits analytically, try the following:
• Direct Substitution
• Factoring/Dividing Out Technique (S.U.A.)
• Rationalize Numerator/Denominator (S.U.A.)
• Other creative (LEGAL) versions of S.U.A.
• Or, you can try…
Calculus: Chapter 1 – Section 1.3
Section 1.3: Evaluation Limits Analytically
The Squeeze/Sandwich Theorem
If h(x) < f(x) < g(x) for all x in an open interval
containing c (except possibly at c), and if
lim h( x)  L  lim g ( x)
x c
Then
lim f ( x)
x c
x c
exists and is equal to L.
Calculus: Chapter 1 – Section 1.3
Section 1.3: Evaluation Limits Analytically
Two Trig Limits (which will be easier to
do in a few months…)
sin x
lim
1
x 0
x
1  cos x
lim
0
x 0
x
Calculus: Chapter 1 – Section 1.3
Section 1.4: Continuity and One-Sided Limits
SECTION 1.4
Continuity and One-Sided Limits
Calculus: Chapter 1 – Section 1.4
Section 1.4: Continuity and One-Sided Limits
What you are about to see
is one the most important
things that you will ever
learn in calculus.
If you forget this, I will
forget you.
Calculus: Chapter 1 – Section 1.4
Section 1.4: Continuity and One-Sided Limits
Definition of Continuity at a Point
A function f is continuous at c if the following three
conditions are met:
1.
f (c) is defined.
2.
lim f ( x)
3.
lim f ( x)  f (c)
x c
exists.
x c
.
DON’T YOU DARE FORGET THIS. EVER.
Calculus: Chapter 1 – Section 1.4
Section 1.4: Continuity and One-Sided Limits
Types of Discontinuity
• Removable: hole in the curve (typically)
 If removable, you can ‘re-define’ the function
to fill the hole
• Non-removable: gap, step, asymptote
Calculus: Chapter 1 – Section 1.4
Section 1.4: Continuity and One-Sided Limits
One-sided Limits
Notation for a one-side limit typically appears as
follows:
• Right-hand Limit:
• Left-hand Limit:
Calculus: Chapter 1 – Section 1.4
lim f ( x)  L
x c
lim f ( x)  L
x c
Section 1.4: Continuity and One-Sided Limits
The Existence of a Limit
Let f be a function and let c be real numbers. The
limit of f(x) as x approaches c is L if and only if
lim f ( x )  L  lim f ( x )
xc
xc
Remember the 5th Grader Definition!
Calculus: Chapter 1 – Section 1.4
Section 1.4: Continuity and One-Sided Limits
Definition of Continuity on Closed Interval
A function f is continuous on [a, b] if it is
continuous on (a, b) and
lim f ( x )  f ( a )
xa
Calculus: Chapter 1 – Section 1.4
and
lim f ( x )  f (b )
x b
Section 1.4: Continuity and One-Sided Limits
Properties of Continuity
If b is a real number and f and g are continuous at
x = c, then following functions are also
continuous at c:
bf
1. Scalar Multiple:
2. Sum/Difference:
3. Product:
f g
fg
4. Quotient:
f
g
Calculus: Chapter 1 – Section 1.4
if
g (c )  0
Section 1.4: Continuity and One-Sided Limits
Intermediate Value Theorem
If f is continuous on the closed interval [a, b] and
k is any number between f(a) and f(b), then
there is at least one number c in [a, b] such
that:
f (c )  k
VERY IMPORTANT THEOREM!!!!!!!
Calculus: Chapter 1 – Section 1.4
Section 1.5: Infinite Limits
SECTION 1.5
Inifinite Limits
Calculus: Chapter 1 – Section 1.5
Section 1.5: Infinite Limits
Inifinite Limits
Let f be a function that is defined at every real
number in some open interval containing c (except
possibly at c itself). The statement:
lim f ( x)  
x c
Mean that for each M > 0 there exists a  > 0 such
that f(x) > M whenever 0  x  c  
*similar definition for limits approaching
Calculus: Chapter 1 – Section 1.5

Section 1.5: Infinite Limits
Vertical Asymptotes
If f(x) approaches infinity (or negative infinity) as
x approaches c from the right or the left, then the
line x = c is a vertical asymptote of the graph of f.
*Be careful with rational functions that can reduce
by dividing out factors
Calculus: Chapter 1 – Section 1.5
Section 1.5: Infinite Limits
Properties of Infinite Limits
Properties of infinite limits are similar to those of
regular limits
See p. 87 for more details.
Calculus: Chapter 1 – Section 1.5