Ministry of Higher and Scientific Education
Erbil Polytechnical University
Khabat Technical Institute
Dept.Of Information Tecnology
Morning-Stage 1
Report About
Limits
Prepared By: - Supervised By: -
Abdulsamad Dashty Mr.Hemn Engineviner
Maawa Firas
Daniel Muhammad
2023-2024
0
Number Names
2 Introduction 1.
3 Example 1 2.
4 Factoring and reducing 3.
5 Infinite Limits 4.
5 Piecewise functions 5.
6-7 Example 2 6.
8 Determining a Limit Graphically 7.
9 Example3 8.
10 Source 9.
1
Introduction
The limit is the value that a function approaches as the input value
approaches the specified amount.
Limits are used to define continuity, derivatives, and integrals.
This booklet is about gifting
Analytical determination of boundaries
There are many ways to determine a limit analytically and they are
usually used in combination.
First, check if the limit can be found by direct substitution. Second,
if direct substitution is one undefined result, factor and subtract the
fraction or multiply by conjugate.
Third, if the second method does not work, we find the left and
right side boundaries.
Take the value of the limit and evaluate the function at this value.
There are two situations where direct substitution will be used,
direct substitution with a numerical value and direct substitution
with infinity
2
Example 1
lim √ 2 x + 9+3 x −1 =¿ ¿
3
x →2 4 x +1
√2(2)3 +9+ 3 ( 2 )−1 = √25+5 = 10
4 ( 2 ) +1 9 9
Limits at infinity are solved by plugging in ∞ or −∞ into the function for the
given variable. When determining limits at infinity, think more about the
trends of the function at infinity rather than the math
x→−∞
lim 2+
( 10
x
2 )
=¿ ¿ ¿
Think about the decimal value of a fraction with a large number in the
denominator. What is the trend? As the denominator gets larger, the fraction
as a whole gets smaller until it ultimately reaches zero. Evaluating using
arbitrary large numbers for infinity will show this trend
10
100 =0.1
10
100,000 =0,0001
Therefore, it can be said that a constant numerator divided by infinity is
equal to zero
2+0=2
x→−∞
lim 2+
( 10
x
2 )
=2+ 0 = 2
3
Factoring and reducing
x−4 0
lim =
x→ 4 √ x−2 0
If x were simply evaluated at 4 as shown in the first method, it would yield a
zero in the denominator; therefore, the slope is undefined. One way to avoid
this is to factor the numerator and denominator if applicable.
lim ¿¿
x→ 4
Reduced by(√ x−2 ¿
lim √ x +2
x→ 4
Now, if x is evaluated at 4, the equation will not yield an undefined slope
√ 4 +2 =4
x−4
lim =4
x→ 4 √ x−2
Piecewise functions
4
1. To find the limit as x approaches 1 from the left side, the first
equation must be used because it defines the function at values less
than and equal to one. Since the equation will not yield an
undefined result, direct substitution can be used
To find the limit as 𝑥𝑥 approaches 1 from the right side, the second
equation must be used because it defines the function at values
greater than one. Since the equation will not yield an undefined
result, direct substitution can be used
The limit of this function does not exist (DNE) because the values
for the left and right sided limits as x approaches 1 yields two
different answers
Infinite Limits
2. . Sometimes when computing limits, an answer of ∞ or −∞ will be
reached, resulting in an infinite limit
Direct substitution will yield a denominator of zero and the
function is already reduced to its simplest form so left and right
sided limits must be used, The graph of this function is shown on
the next page in Figure 2.27.
Example 2
5
1. f ( x )=
{−2¿ √ x−1
x+ 4 x ≤1
x >1
lim ¿
−¿
x→ 1 (− 2 x +4 ) =−2( 1) + 4=2 ¿
lim ¿
x→ 1
+¿
√ x−1= √( 1) −1=0 ¿
f ( x )=
{−2¿ √ x−1
x+ 4 x ≤1
x >1
=DNE
X−2
2. lim
X→1 ( X−1 )2 ( X−3)
To evaluate the left and right sided limits,
evaluate the function for values very close
to the limit. Think about the decimal value
of a fraction with a small number in the
denominator. What is the trend? As the
denominator gets smaller the fraction as a
whole gets larger until it ultimately reaches
infinity. Evaluating using numbers close to
the limit will show this trend:
To evaluate the limit as x approached one from the right, try evaluating with
a number slightly greater than one.
lim ¿
+¿ ( 1.01) −2
x→ 1 f ( 1.01) = ¿
¿¿
−0.99
¿ ≈ 4 , 97 5
(0.0001)(−1.99)
Example 2
Now, try evaluating with a number closer to 1
6
lim ¿
+¿ ( 1. 000001 ) −2
x→ 1 f ( 1.0 00001) = ¿
¿¿
To evaluate the limit as 𝑥𝑥 approaches one from the left, try evaluating with
a number slightly less than 1.
lim ¿
+¿ ( 0.9 ) −2
x→ 1 f ( 0.9) = ¿
¿¿
Now, try evaluating with a number closer to 1
lim ¿
+¿ ( 0.999999 ) −2
x→ 1 f ( 0.9) = ¿
¿¿
As the denominator gets closer to zero, the function becomes larger and
gets closer to infinity. Therefore: lim ¿ ∞ since both the left and the right
x →1
side of the limit approach infinity
Determining a Limit Graphically
7
There are two types of conditions to be aware of when determining limits
graphically, areas where a function is continuous and areas where a
function is discontinuous. Suppose the following graph is used to
determine various limits
Continuous Limits
A function is continuous if the graph contains no abrupt changes in x and y
values (i.e., no holes, asymptotes, jumps, or breaks).
Using Figure 2.45, look at the limit as x approaches 4. The graph does not
have any holes or asymptotes at x = 4, therefore a limit exists and is equal to
the y value of the function
lim f ( x )=2
x→ 4
Discontinuous Limits A function is discontinuous if there is an abrupt
change in x and y values (i.e. holes, asymptotes, jumps, and/or breaks
exist)
Example 3
8
The graph as x approaches 1 is discontinuous because there is a hole at x = 1
and therefore no value. However, evaluating the left and right sided limits
will determine if a limit still exists at x = 1. Looking at the graph, as x
approaches 1 from the left side the function approaches 3.
lim ¿
−¿
x→ 1 f ( x ) =3 ¿
Looking at the graph as x approaches 1 from the right side, the function
approaches 3
lim ¿
+¿
x→ 1 f ( x ) =3 ¿
Since both the left-sided and right-sided limits have the same value, a
limit exists for this function.
lim f ( x )=3
x →1
Source
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https://math.libretexts.org/Bookshelves/Precalculus/
Precalculus_1e_(OpenStax)/12%3A_Introduction_to_Calculus/
12.01%3A_Finding_Limits_-_Numerical_and_Graphical_Approaches
https://www.geeksforgeeks.org/limits-by-direct-substitution/
https://math.washington.edu/~conroy/2013/m120-aut2013/conjugate.pdf
https://sites.millersville.edu/bikenaga/calculus1/left-and-right-limits/left-
and-right-limits.html
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