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In mathematics, limits refer to the values that a function approaches as its input approaches a certain value or goes to infinity. It is a fundamental concept in calculus, and it plays a crucial role in understanding the behavior of functions and solving many mathematical problems.
The idea of limits was first introduced in the 17th century by mathematicians like John Wallis, Isaac Barrow, and James Gregory. However, it was not until the 19th century that the concept of limits was fully developed and formalized by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass.
In this article, we will discuss the limits, properties, and some basic rules, and also explain the topic with help of an example.
Definition of Limit Calculus
Let f(x) be a function defined on an open interval containing a point c, except possibly at c itself. Then, the limit of f(x) as x approaches c is L, denoted by:
lim x → c f(x) = L
if for every positive number ε, there exists a positive number δ such that if 0 < |x - c| < δ, then |f(x) - L| < ε.
This definition means that we can make f(x) arbitrarily close to L by taking x sufficiently close to c, but not equal to c itself. In other words, we can make the difference between f(x) and L as small as we like by choosing x close enough to c.
What is a limit?
In mathematics, limits are distinct real numbers. Consider the limit of a real-valued function "f" and a real number "c," which is typically described as lim x→ c f (x) = L. It is written as "the limit of f of x, as x approaches c equals L".
The "lim" indicates the limit, and the right arrow indicates that function f(x) approaches the limit L as x approaches c.
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6 f(x)
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Properties of limits
There are several properties of limits that are important in calculus and other areas of mathematics. These properties can be used to simplify the evaluation of limits and to prove more advanced results. Here are some of the most important properties of limits:
Sum and difference
If the limits of two functions f(x) and g(x) exist as x approaches b, then the limit of their sum or difference also exists, and it is equivalent to the difference or sum of their respective limits:
lim x → b [f(x) ± g(x)] = lim x → b f(x) ± lim x → b g(x)
Product
If the limits of two functions f(x) and g(x) exist as x approaches c, then the limit of their product also exists, and it is equal to the product of their limits:
lim x → b [f(x) × g(x)] = lim x → b f(x) × lim x → b g(x)
Quotient
If the limits of two functions f(x) and g(x) exist as x approaches b, and g(x) is not equal to zero for x near b, then the limit of their quotient also exists, and it is equal to the quotient of their limits:
lim x → b [f(x) / g(x)] = [lim x → b f(x)] / [lim x → b g(x)]
Power
If the limit of a function f(x) exists as x approaches b, then the limit of its nth power also exists, and it is equal to the nth power of its limit:
lim x → b [f(x)^n] = [lim x → b f(x)] ^n
Constant
If c is a constant, then the limit of c as x approaches any value is c:
lim_x → b (c) = c
Identity
If f(x) = x, then the limit of f(x) as x approaches any value is that value:
lim x → b (x) = b
Exponential
If the limit of a function f(x) exists as x approaches a, and c is a constant, then the limit of c^f(x) also exists, and it is equal to c raised to the limit of f(x):
lim_x → a c^f(x) = c^ [lim_x → a f(x)]
These basic properties of limits are very useful in calculus to evaluate limits to get the numerical result of a function at a specific point.
Examples of Limits
Example 1:
Calculate the limit of limx→4[(x+3) (x-2)/x-2]
Solution:
Given limx→4[(x+3) (x-2)/x-2]
Step1:
limx→4[(x+3) (x-2)/x-2] = [ limx→4(x+3) limx→4(x-2)/ limx→4(x-2)]
Step2:
Now, simplify
limx→4[(x+3) (x-2)/x-2] = [(4+3) (4-3)/ (4-2)]
= [(7)(1)/ (2)]
= 7/2
Example 2:
Evaluate limx→∞[3x2+3/x2-5x-1]
Solution:
Given
limx→∞[3x2+3/x2-5x-1]
Step 1:
Factor the largest power of x in the numerator from each and the largest power of x in the denominator from each term.
limx→∞[3x2+3/x2-5x-1] = limx→∞[x2(3+3/x2)/ x2 (1 -5/x-1/ x2)]
Step 2:
limx→∞[3x2+3/x2-5x-1] = limx→∞ [((3+3/x2)/ (1 –(5/x)-1/ x2)]
Now we put the limit
= [(3+3/ (∞)2)/ (1-5/∞ –1/ (∞)2]
= (3+0/1-0-0)
limx→∞[3x2+3/x2-5x-1] = 3
Summary
In this article, we have discussed the introduction of the limit, the basic definition of limit, properties, and rules will be discussed and also with the help of an example topic will be explained. After studying this article everyone can easily defend this topic.