Lecture 2- BASIC LIMITS and INDETERMINATE FORM.pptx

Basic Limits and the Indeterminate
Form

Definition of Limits
• If f(x) is a function and 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. This limit is written
mathematically as
and is read as “the limit of f(x) as x approaches c is L”

• To illustrate this
definition, let’s take
for example the
function
• whose graph is
shown

• For values other than x = 1,
you can use standard curve-
sketching techniques. At x =
1, however, it is not clear
what to expect.

• To get an idea of the
behavior of the graph of
f(x) near x = 1, we can use
two sets of x-values—one
set that approaches 1 from
the left and one set that
approaches 1 from the right
•

Hence, we could say that f(x) approaches the value 3 as the
value of x approaches 1 from either the left or the right.

Therefore, the limit of f(x) as x approaches 1 is 3.

Estimating a Limit Numerically

• Example 1:
• For the function
• discuss the behavior of the values of f(x) when x gets closer to 2 using
table

• Therefore,

• Example 2:
• Find the limit of
as x approaches zero

• Example 2:
• Evaluate the function at
several x-values near 0 and
use the results to estimate
the limit
• *As in the graph, we could
see that f(0) is undefined.
For this reason, we cannot
find the limit by finding
f(0) as in Example 1

• Example 2:
• To estimate the limit of
f(x) as x approaches 0,
a list of several values
of x near zero from left
and right would help.

• Example 3:
• For the function
discuss the behavior of
the values of f(x) when x
is closer to 2.
Does the limit exist?
2
2
)
(



x
x
x
f

* This function is not defined when x = 2.
* The limit does not exist because the limit on the
left and the limit on the right are not the same.
represents the limit on the left of 2
• represents the limit on the right of 2
x 0 1 1.9 1.99 2 2.001 2.0
1
2.1 2.5
f (x) -1 -1 -1 -1 ? 1 1 1 1

■ We write and call K the limit from the left (or left-hand
limit) if
f(x) is close to K whenever x is close to c, but to the left
of c on the real number line.
■ We write and call L the limit from the right (or right-hand
limit) if f(x) is close to L whenever x is close to c, but to the
right of c on the real number line.
■ In order for a limit to exist, the limit from the left and the
limit from the right must exist and be equal.

Estimating a Limit Using Graph

Example 1:
From the given graph of f(x),
answer the following
a. f(0)
¿ 0
¿ 0
¿ 0
¿ 0

Example 1:
e. f(1)
¿ 1
¿ 2
¿ Does not exist
¿NotDefined

Example 1:
i. f(3)
¿ 3
¿ 3
¿ 3
¿NotDefined

Evaluating Limits Analytically

Lecture 2- BASIC LIMITS and INDETERMINATE FORM.pptx

Example 1:
Try lim (x4
+ 3x – 2)
X-1
If you don’t get -4, try again

Example 2:
Property 8
Try lim
X-1
2
2 2

x If you don’t get 2, try again

Example 3:
Note that this is a rational function with a
nonzero denominator at x = -2
2
lim 4
1 
 x
x
Try
x
If you don’t get 1/3, try again

Example 4:







12
3
2
)
(
x
x
x
f
If x < 5
If x > 5
Find:
a. f(5)
This is called “Piecewise Function”

Example 4:







12
3
2
)
(
x
x
x
f
If x < 5
If x > 5
Solution:
b.

Example 4:







12
3
2
)
(
x
x
x
f
If x < 5
If x > 5
Solution:

Example 1: Use algebraic and/or graphical techniques
to analyze each of the following indeterminate forms

Example 2: Evaluate the limit
note: a3
- b3
= (a - b)(a2
+ ab + b2
)

SEATWORK:
2.
3.
lim
𝑥→1
(1+3 𝑥)2

Lecture 2- BASIC LIMITS and INDETERMINATE FORM.pptx

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