CALCULUS I

Research Assistant Dr. Gökçe ÇAKMAK

Week 3
Limits of Functions

Let f(x) be defined on an open interval about 𝑥0 , except possibly at 𝑥0 itself. If

f(x) gets arbitrarily close to L (as close to L as we like) for all x sufficiently close
to 𝑥0 we say that f approaches the limit L as x approaches and we write

lim 𝑓(𝑥) = 𝐿
𝑥→𝑥0
How does the function

𝑥2 − 1
𝑓 𝑥 =
𝑥−1
behave near x=1?
The Limit Value Does Not Depend on How the Function
Is Defined at 𝒙𝟎
✓ Sometimes lim 𝑓(𝑥) can be evaluated by calculating 𝑓(𝑥0 ).
𝑥→𝑥0

✓ If f is the identity function f(x)=x then for any value of 𝑥0 ,

lim 𝑓(𝑥) = lim 𝑥 = 𝑥0 .

𝑥→𝑥0 𝑥→𝑥0

✓ If f is the constant function f(x)=k (function with the constant value k), then
for any value of 𝑥0

lim 𝑓(𝑥) = lim 𝑘 = k.

𝑥→𝑥0 𝑥→𝑥0
Evaluate the limit or explain why it does not exist.

𝑥 2 − 6𝑥 + 9
lim
𝑥→3 𝑥2 − 9

4+ℎ−2
lim
ℎ→0 ℎ

𝑥, 𝑥 ≠ 2
𝑔 𝑥 =ቊ . lim 𝑔(𝑥) =?
1, 𝑥 = 2 𝑥→2
EXISTENCE OF lim 𝑓(𝑥) DOES NOT REQUIRE THAT
𝑥→𝑥0

f(𝑥0 ) EXIST AND DOES NOT DEPEND ON f(𝑥0 ) EVEN IF

f(𝑥0 ) DOES EXIST !!!!!
A Function May Fail to Have a Limit at a Point in Its
Domain

It grows too large to It oscillates too

It jumps
have a limit much to have a limit
One-Sided Limits

If f(x) is defined on some interval (b, a) extending to the left of x=a, and if
we can ensure that f(x) is as close as we want to L by taking x to the left of a
and close enough to a, then we say f(x) has left limit L at x=a, and we write

lim 𝑓(𝑥) = 𝐿
𝑥→𝑎−
If f(x) is defined on some interval (a, b) extending to the right of x=a, and if
we can ensure that f(x) is as close as we want to L by taking x to the right of a
and close enough to a, then we say f (x) has right limit L at x=a, and we write

lim 𝑓(𝑥) = 𝐿
𝑥→𝑎+
Relationship between one-sided and two-sided limits:

A function f(x) has limit L at x = a if and only if it has both left and right limits
there and these one-sided limits are both equal to L:

lim 𝑓(𝑥) = 𝐿 ⟺ lim− 𝑓 𝑥 = lim+ 𝑓 𝑥 = 𝐿

𝑥→𝑎 𝑥→𝑎 𝑥→𝑎
Evaluate the limit or explain why it does not exist.

|𝑥 − 2|
lim
𝑥→2 𝑥 − 2

𝑥2 − 4
lim−
𝑥→2 |𝑥 + 2|
Rules for Calculating Limits

If lim 𝑓(𝑥) = 𝐿, lim 𝑔(𝑥) = 𝑀, and k is a constant, then

𝑥→𝑎 𝑥→𝑎

Limit of a sum: lim [𝑓(𝑥) + 𝑔(𝑥)] = 𝐿 + 𝑀

𝑥→𝑎

Limit of a difference: lim 𝑓 𝑥 − 𝑔 𝑥 =𝐿−𝑀

𝑥→𝑎

Limit of a product: lim 𝑓 𝑥 𝑔 𝑥 = 𝐿𝑀

𝑥→𝑎

Limit of a multiple: lim 𝑘𝑓(𝑥) = 𝑘𝐿

𝑥→𝑎
𝑓(𝑥) 𝐿
Limit of a quotient: lim = , 𝑖𝑓 𝑀 ≠ 0.
𝑥→𝑎 𝑔(𝑥) 𝑀
 𝑓 𝑥 −5
If lim = 3 , find lim 𝑓(𝑥) .
𝑥→2 𝑥−2 𝑥→2
Rules for Calculating Limits (contd.)

Limit of a power: If m is an integer and n is a positive integer, then

lim 𝑓 𝑥 𝑚/𝑛 = 𝐿𝑚/𝑛

𝑥→𝑎

provided L>0 if n is even and 𝐿 ≠ 0 if m<0.

Order is preserved: If 𝑓 𝑥 ≤ 𝑔(𝑥) on an interval containing a in its interior,
then 𝐿 ≤ 𝑀.

Rules are also valid for right limits and left limits, except the last one is valid
under the assumption that 𝑓 𝑥 ≤ 𝑔(𝑥) on an open interval extending in the
appropriate direction from a.
Limits of Polynomials and Rational Functions

If P(x) is a polynomial and a is any real number, then

lim 𝑃(𝑥) = 𝑃 𝑎
𝑥→𝑎

If P(x) and Q(x) are polynomials and 𝑄 𝑎 ≠ 0, then

𝑃(𝑥) 𝑃(𝑎)
lim =
𝑥→𝑎 𝑄(𝑥) 𝑄(𝑎)
The Sandwich/Squeeze Theorem

Suppose that 𝑔 𝑥 ≤ 𝑓 𝑥 ≤ ℎ(𝑥) for all

x in some open interval containing c,
except possibly at itself. Suppose also
that

lim 𝑔(𝑥) = lim ℎ(𝑥) = 𝐿.

𝑥→𝑐 𝑥→𝑐
Then

lim 𝑓(𝑥) = 𝐿
𝑥→𝑐
Limits at Infinity

«f(x) approaches 0 as x approaches infinity.»

«f(x) approaches 0 as x approaches negative infinity.»
The line y=0 (or x-axis) is called horizontal asymptote of the graph. In general,
if a curve approaches a straight line as it recedes very far away from the origin,
that line is called an asymptote of the curve.
Limits at infinity and negative infinity (informal definition)

If the function f is defined on an interval (𝑎, ∞) and if we can ensure that f(x)
is as close as we want to the number L by taking x large enough, then we say
that f(x) approaches the limit L as x approaches infinity, and we write

lim 𝑓(𝑥) = 𝐿
𝑥→∞
If f is defined on an interval (−∞, 𝑏) and if we can ensure that f(x) is as close
as we want to the number M by taking x negative and large enough in absolute
value, then we say that f(x) approaches the limit M as x approaches negative
infinity, and we write

lim 𝑓(𝑥) = 𝑀.
𝑥→−∞
Limits at ±∞ for rational functions

Let 𝑃𝑚 𝑥 = 𝑎𝑚 𝑥 𝑚 + ⋯ + 𝑎0 and 𝑄𝑛 𝑥 = 𝑏𝑛 𝑥 𝑛 + ⋯ + 𝑏0 be polynomials of

degree m and n respectively, so that 𝑎𝑚 ≠ 0. Then

𝑃𝑚 (𝑥)
lim
𝑥→±∞ 𝑄𝑛 (𝑥)

a. equals to zero if m<n.

𝑎𝑚
b. equals if m=n.
𝑏𝑛

c. does not exist if m>n.

Examples

𝑥2 + 3
lim 3
𝑥→−∞ 𝑥 + 2

𝑥 2 + 𝑠𝑖𝑛𝑥
lim 2
𝑥→∞ 𝑥 + 𝑐𝑜𝑠𝑥

2𝑥 − 1
lim
𝑥→−∞ 3𝑥 2 + 𝑥 + 1
Infinite Limits

As x approaches 0 from either side, the values

of f(x) are positive and grow larger and larger,
so the limit of f(x) as x approaches 0 does
not exist.
f(x) approaches infinity as x approaches zero.

The y-axis (or x=0) is a vertical asymptote of the

graph.
One-Sided Infinite Limits

As x approaches 0 from the right, the values

of f(x) become larger and larger positive
numbers, and we say that f has right-hand
limit infinity at x=0.

The values of f(x) become larger and larger

negative numbers as x approaches 0 from
the left, so f has left-hand limit negative
infinity at x=0.
examples

1
lim
𝑥→3 3 − 𝑥

1
lim 2
𝑥→3 3 − 𝑥

1
lim+
𝑥→1 |𝑥 − 1|

1
lim
𝑥→1− |𝑥 − 1|
Continuity at a Point

A point P in the domain of a function is called an interior point of the domain if it

belongs to some open interval contained in the domain. If it is not an interior
point, then P is called an endpoint of the domain.
Continuity at an interior point

We say that a function f is continuous at an interior point c of its domain if

lim 𝑓(𝑥) = 𝑓(𝑐)

𝑥→𝑐

If either lim 𝑓(𝑥) fails to exist or it exists but is not equal to f(c), then we
𝑥→𝑐
will say that f is discontinuous at c.
f is continuous at c lim 𝑓(𝑥) ≠ 𝑓 𝑐 lim 𝑓(𝑥) 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
𝑥→𝑐 𝑥→𝑐
Right and left continuity

We say that f is right continuous at c if

lim+ 𝑓(𝑥) = 𝑓 𝑐
𝑥→𝑐

We say that f is left continuous at c if

lim 𝑓(𝑥) = 𝑓(𝑐)

𝑥→𝑐 −
Function f is continuous at c if and only if it is both right
continuous and left continuous at c.
Continuity at an endpoint

We say that f is continuous at a left endpoint c of its domain

if it is right continuous there.

We say that f is continuous at a right endpoint c of its

domain if it is left continuous there.
Continuity on an interval

We say that function f is continuous on the interval I if it is

continuous at each point of I.

In particular, we will say that f is a continuous function if f is

continuous at every point of its domain.
If the functions f and g are both defined on an interval containing c and both are
continuous at c, then the following functions are also continuous at c:
1. the sum f + g and the difference f - g;
2. the product fg;
3. the constant multiple kf, where k is any number;
4. the quotient f/g (provided g(c) not equal to 0);
5. the nth root 𝑓 𝑥 1/𝑛 provided f(c) > 0 if n is even.
Composites of continuous functions are continuous

If f (g(x)) is defined on an interval containing c, and if f is continuous at L and

lim 𝑔 𝑥 = 𝐿 , then
𝑥→𝑐

lim 𝑓 𝑔 𝑥 = 𝑓(𝐿) = 𝑓(lim 𝑔(𝑥))

𝑥→𝑐 𝑥→𝑐

In particular, if g is continuous at c (so L=g(c)), then the composition 𝑓 ∘ 𝑔 is

continuous at c:

lim 𝑓 𝑔 𝑥 = 𝑓(𝑔(𝑐))
𝑥→𝑐
If f(c) is not defined, but lim 𝑓(𝑥) = 𝐿 exists, we can define a new function F(x)
𝑥→𝑐
by

𝑓 𝑥 , 𝑖𝑓 𝑥 ∈ 𝐷𝑜𝑚(𝑓)
𝐹 𝑥 =ቊ
𝐿, 𝑖𝑓 𝑥 = 𝑐

F(x) is continuous at x=c. It is called the continuous extension of f (x) to x=c.

For rational functions f, continuous extensions are usually found by cancelling

common factors.
If a function f is undefined or discontinuous at a point a but can be (re)defined
at that single point so that it becomes continuous there, then we say that f has
a removable discontinuity at a.
How should the given function be defined at the given point to be continuous
there? Give a formula for the continuous extension to that point.

𝑡 2 − 5𝑡 + 6
𝑎𝑡 3
𝑡2 − 𝑡 − 6
The Max-Min Theorem

If f (x) is continuous on the closed, finite interval [a, b], then there exist numbers
p and q in [a, b] such that for all x in [a, b], 𝑓 𝑝 ≤ 𝑓 𝑥 ≤ 𝑓(𝑞).
Thus f has the absolute minimum value m = f (p), taken on at the point p, and
the absolute maximum value M = f (q), taken on at the point q.
Not bounded Bounded Has no max value Bounded
has no max or min Has no max or min Has no max or min
The Intermediate-Value Theorem

If f (x) is continuous on the interval [a, b] and if s is a number between f

(a) and f (b), then there exists a number c in [a, b] such that f (c) = s.
Show that the equation 𝑥 3 − 𝑥 − 1 = 0 has a solution on the interval [1,2].