In Calculus, limit provide a way to study and analyze the
behavior of functions as the input approaches a particular value. The concept
of limits allows us to investigate questions such as "What is the value of
a function at a specific point?" / "Does a function have a well-defined
value as the input gets arbitrarily close to a certain value?"
Evaluating limits involves various techniques, such as
direct substitution, factoring, rationalization, and the use of special
trigonometric identities. In this article, we will discuss the definition of
limit, its rule, and types with solved examples.
What is the limit in Mathematics?
In mathematics, a limit is a fundamental concept that
describes the behavior of a function as the input values approach a particular
value. It represents the value that a function "approaches" or gets
arbitrarily close to, without actually reaching it.
Formally, the limit of a function f(x) as x approaches a
value c is denoted as follows:
lim_{(x → c)}
f(x) = L
This notation indicates that as x gets arbitrarily close to
c (from both sides), the corresponding values of f(x) get arbitrarily close to
L. The limit L may exist and be a well-defined value, or it may not exist
depending on the behavior
of the function.
Rules of Limit
There are several important rules/properties associated with
limits that are commonly used in calculus. Here are some of the key rules of
limits:
Rules |
Function |
Output |
Limit of a Constant |
lim_{(x→a) }c = |
c |
Limit of the
Identity Function |
Lim_{(x→a)} x = |
a |
Sum and
Difference Rule |
lim_{(x→a)} [f(x) ± g(x)] = |
L ± M |
Product Rule |
lim_{(x→a)} [f(x) × g(x)] = |
L ×
M |
Quotient
Rule |
lim_{(x→a)} [f(x) / g(x)] = |
L / M |
Power Rule |
lim_{(x→a)} [f(x)^{n}] = if lim_{(x→a)} f(x) = |
L^{n} |
Exponential |
lim_{(x→a)} e^{x} = |
e^{a} |
L'HÃ´pital's rule is a well-known technique for finding
limits involving indeterminate forms, where both the numerator and denominator
approach zero or infinity. To deal with this kind of problems, you can get help
from an L’hopital’s rule calculator.
Types of limits
Limits can take on different forms based on the behavior of
the function as the input approaches the given value. Here are a few common
types of limits:
·
Finite Limits: A finite limit exists when
the function approaches a specific finite value as the input approaches a
particular point.
·
Infinite Limits: An infinite limit occurs
when the function grows or diminishes without bounds as the input approaches a
certain value. It can approach positive infinity (∞), negative infinity (-∞),
or both.
·
Limits at Infinity: Limits at infinity
describe the behavior of a function as the input approaches positive or
negative infinity. The limit can be a finite value, infinity, or may not exist.
The limit has broad applications across mathematics,
science, and engineering. It is used to analyze rates of change, determine
critical points, solve optimization problems, define probability distributions,
analyze algorithms, and study the behavior of physical systems.
How to Find Limits?
You can take assistance from an online limit calculator
to find limits with steps without involving in lengthy calculations. Here are a
few examples to find limits manually with the help of rules of limit.
Example 1:
lim_{x→}-_{3}(5x^{3} - 10x^{2}
+15x - 20)
lim_{ x→}-_{3}=?
Solution
Step 1:
In the first step, we
apply the given limit to the function.
= lim_{x→}-_{3}(5x^{3}) - lim_{x→}-_{3}(10x^{2})
+ lim_{x→}-_{3}(15x) - lim_{x→}-_{3}(20)
Step 2:
In the second step, we take out the limit function constant
value.
= 5lim_{x→}-_{3}(x^{3}) -10 lim_{x→}-_{3}(x^{2})
+15 lim_{x→}-_{3}(x) - lim_{x→}-_{3}(20)
Step 3:
Apply the value of the limit in the given function. Here,
the value of the limit is x=1
Apply the limit value x=1 we get.
lim_{x→}-_{3}(5x^{3} - 10x^{2}+
15x - 20) = 5(-3)^{3} -10(-3)^{2} +15(-3) -20
lim_{x→}-_{3}(5x^{3} - 10x^{2}+
15x - 20) = -167
Example 2:
lim_{x→-2}[(10x^{2} +15x - 20)/3x^{2}+1]
lim_{x→-2}=?
Solution
we can find the limit of the following function by using
step-by-step.
Step 1:
Apply the given limit to the function
lim_{x→-2}[(10x^{2} +15x - 20)/(3x^{2}+1)]
= lim_{x→-2}[(10x^{2} +15x - 20)/ lim_{x→-2}(3x^{2}+1)
step 2:
The coefficient of the function get out of the limit and
apply limit to the all function.
lim_{x→-2}[(10x^{2} +15x - 20)/(3x^{2}+1)]
= [10 lim_{x→-2 }(x^{2}) +15 lim_{x→-2 (}x) -20 lim_{x→-2
}(1)]/ [ lim_{x→-2}(x^{2}) +1 lim_{x→-2}(1)]
step 3:
Now put the limit value
lim_{x→-2}[(10x^{2} +15x - 20)/(3x^{2}+1)]
=10(-2)^{2}+15(-2)-20(1)/ (-2)^{2}+1
step 4:
simplify the question
=10(4)-30-20/4+1
lim_{x→-2}[(10x^{2} +15x - 20)/(3x^{2}+1)]
=-10/5 =-2
FAQs
Question 1:
What does the mean of limit existing?
Answer:
For a limit to exist, the function's output values must
approach a specific value as the input values get arbitrarily close to the
given value. If this occurs, the limit is said to exist
Question 2:
How do you evaluate
limits algebraically?
Answer:
To evaluate limits algebraically, you can apply various
limit rules and algebraic manipulations. These include the sum, difference,
product, quotient, and power rules mentioned earlier. Additionally, factoring,
rationalizing, and simplifying expressions can often help in evaluating limits.
Question 3:
How is a limit denoted?
Answer:
A limit is denoted
using mathematical notation. The general notation for the limit of a function
f(x) as x approaches a particular value a is written as:
lim_{(x→a)}
f(x).
Conclusion
In this article, we have discussed the definition of limit, along
with its rules, types, and examples. Now you can seek proper guidance about
limits from this post. You have to solve several problems by hand to master the
calculations of limits.