What is a Limit?
Limits examples are one of the most difficult concepts in Mathematics according to many students. However, through easier understanding and continued practice, students can become thorough with the concepts of what is limits in maths, the limit of a function example, limits definition and properties of limits. Limit math is one of the most important concepts in Calculus. Calculus is a branch of mathematics that deals with the calculations related to continuously changing quantities. The math limit formula can be defined as the value that a function returns as an output for the given input values.
What are Limits & Limits Formula in Maths?
Limits math is very important in calculus. It is one of the basic prerequisites to understand other concepts in Calculus such as continuity, differentiation, integration limit formula, etc. Most of the time, math limit formulas are the representation of the behaviour of the function at a specific point. Hence, the concept of limits is used to analyze the function. The mathematical concept of the Limit of a topological net generalizes the limit of a sequence and hence relates limits math to the theory category. Integrals in general are classified into definite and indefinite integrals. The upper and lower limits are specified in the case of the definite integration limit formula. However, indefinite integration limit formulas are defined without specified limits and hence have an arbitrary constant after integration. The subsequent sections elaborate a brief overview of various concepts involved in a better understanding of math limit formulas.
Limits formula:- Let y = f(x) as a function of x. If at a point x = a, f(x) takes indeterminate form, then we can consider the values of the function which is very near to a. If these values tend to some definite unique number as x tends to a, then that obtained unique number is called the limit of f(x) at x = a.
Limits Maths
The limit of a real-valued function ‘f’ with respect to the variable ‘x’ can be defined as:
\[\lim_{x\rightarrow p}f(x)=L\]
In the above equation, the word ‘lim’ refers to the limit.
It generally describes that the real-valued function f(x) tends to attain the limit ‘L’ as ‘x’ tends to ‘p’ and is denoted by a right arrow.
We can read this as: “the limit of any given function ‘f’ of ‘x’ as ‘x’ approaches ‘p’ is equal to ‘L’”.
What are the Properties or Laws of Limits?
The Properties of Limits Are as Follows:
The Notation of a Limit
The limit of a function is denoted as f (x) → L as x → p or in the limit notation as:
\[\lim_{x\rightarrow p}f(x)=L\]
Let us assume that there exists \[\lim_{x\rightarrow p}f(x)\], \[\lim_{x\rightarrow p}g(x)\], \[\lim_{x\rightarrow p}f_1(x)\]……..\[\lim_{x\rightarrow p}f_n(x)\]. This assumption is made to explain the other properties of limits.
The Sum Rule
The sum rule states that the sum of the individual limits of any two functions is equal to the limit of the sum of those functions.
\[\lim_{x\rightarrow p}f(x)\] + \[\lim_{x\rightarrow p}g(x)\] =\[\lim_{x\rightarrow p}\mid f(x)+g(x)\mid \]
The Extended Sum Rule
The extended sum rule is the same as the sum rule. However, it is defined for the limits of more than two functions.
\[\lim_{x\rightarrow p}f_1(x)\] + \[\lim_{x\rightarrow p}f_2(x)\] +........\[\lim_{x\rightarrow p}f_n(x)\] =\[\lim_{x\rightarrow p}\mid f_1(x)+f_2(x)+.......f_n(x)\mid \]
The Constant Function Rule
The constant function rule states that the limit of a constant function is equal to a constant.
The Constant Multiple Rule
The limit of a function, when multiplied by a constant value, is equal to the constant multiplied by the limit of the function.
\[\lim_{x\rightarrow p}kf(x)\] = k \[\lim_{x\rightarrow p}f(x)\]
The Product Rule
The product rule states that the product of the limits of two individual functions is equal to the limit of the product of the functions.
\[\lim_{x\rightarrow p}f(x)\ast \lim_{x\rightarrow p}g(x)=\lim_{x\rightarrow p}\mid f(x)\ast g(x)\mid \]
The Extended Product Rule
The extended product rule is the same as the product rule. However, more than two functions are taken into consideration.
\[\lim_{x\rightarrow p}f_1(x)\ast \lim_{x\rightarrow p}f_2(x)\ast ......\lim_{x\rightarrow p}f_n(x)=\lim_{x\rightarrow p}\mid \lim_{x\rightarrow p}f_1(x)\ast \lim_{x\rightarrow p}f_2(x)\ast ......\lim_{x\rightarrow p}f_n(x)\mid \]
The Quotient Rule
The quotient of individual limits of two functions when the limit of the denominator is not equal to zero is equal to the limit of the quotient of the two functions where the denominator function is not equal to zero.
\[\frac{\displaystyle\lim\limits_{x\rightarrow p}f(x)}{\displaystyle\lim\limits_{x\rightarrow p}g(x)}=\lim_{x\rightarrow p}\frac{f(x)}{g(x)}\]
The Power Rule
The power rule of limits is Mathematically stated as follows:
\[\lim_{x\rightarrow p}{\mid f(x)\mid }^n={\lim_{x\rightarrow p}f(x)^n}\]
where ‘n’ is known to be any integer
Similarly, when the powers are fractions, the power rule can be stated as:
\[\lim_{x\rightarrow p}\sqrt{\mid f(x)\mid }=\sqrt{\lim_{x\rightarrow p}f(x)}\]
Important Limit Formula List in Limits
Math Limit Formula List
Above is the limit formula list.
If the Limits of a Function are given in a Quotient Form, How can I evaluate It?
If the limit of a function is given in a quotient form, you can evaluate the expression by the factoring method. The steps to follow for the same are
Reduce the numerator and denominator to its factors completely.
Now simplify the expression, by dividing the numerator and the denominator to any factor which is common to both.
Now, you may evaluate the resulting limit, keeping the correct domain in mind.
How can I evaluate the Limits of a Quotient?
You can very easily evaluate the limit of a quotient by predicting the LCD of the expression in question. The steps you can take for such evaluation are
Predict the LCD of the two terms in the numerator for the denominator.
Convert both the fractions in the numerator such that they have LCD as their denominator.
Multiply both the denominator as well as the numerator with the LCD.
Now apply the distributive property of limits.
Simplify the resulting expression and factor the numerator.
Cancel the like fractions present in the resulting expression.
Evaluate the expression keeping the limits of x in the formula.
How can I Evaluate the Limit of a Function that Contains a Root?
It is very easy to evaluate the limit of a function containing a root by using the conjugate method. You can apply this method by simply following the steps stated below:
Directly evaluate the quotient if it is not given in the indeterminate form, that is, (0/0).
You can also use the least common denominator (LCD) method to convert the two quotients in question to a single quotient by taking their sum or difference.
If the numerator in question also contains a root, then rationalize the numerator and then multiply the numerator and the denominator with the conjugate of the numerator.
Now simplify the resulting expression and evaluate the resulting limit.
How can I evaluate the Limits, if a Quotient with Absolute Values is given?
You can easily evaluate the limits of an expression containing a quotient with absolute values. Follow the steps given below to solve problems of such kind:
Find the least common denominator (LCD) of the fraction in question.
If you cannot find the limit then choose various values that are close to the undefined function on either side of the input.
Now you can use numeric evidence to analyze the limits on both sides.
Limits of a Function - At A Glance
The various properties of limits are used to perform operations on the limits of the function and not on the function itself.
You can predict the limit of a polynomial function by calculating the sum of the individual terms.
The limit of a function raised to a power is equal to the power of the limit of the function, commonly known as the direct substitution method.
The root of the limit of the function is equal to the limit of the root of a function.
Writing the quotient in factored form and then simplifying can help in finding the limit of the function expressed as a quotient.
One can find the limit of a complex function by finding its LCD.
The conjugate method can be used for finding the limit of a function containing a root.
The factoring method can also be used for finding the limit of some functions
Numeric evidence or piecewise setting can help in evaluating the limit of a quotient containing absolute values.
Limits Examples and Solutions
Evaluate \[\lim_{x\rightarrow 2}\](1x3 - 3x2 + 6x -3)
Ans:
\[\lim_{x\rightarrow 2}\](1x3 - 3x2 + 6x -3)
= \[\lim_{x\rightarrow 2}\](1x3) - \[\lim_{x\rightarrow 2}\](3x2) + \[\lim_{x\rightarrow 2}\](6x) - \[\lim_{x\rightarrow 2}\](3)
= 1\[\lim_{x\rightarrow 2}\](x3) - 3\[\lim_{x\rightarrow 2}\](x2) + 6\[\lim_{x\rightarrow 2}\](x) - (3)
= 1(2)3 - 3(2)2 +6(2) -3
= 1 x 8 - 3 x 4 + 12 - 3
= 8 - 12 + 9
= 17-12
= 5
FAQs on Limits Formula
1. What is the Formula of Limits?
What is a Limit? Let y = f(x) as a function of x. If at a point x = a, f(x) takes indeterminate form, then we can consider the values of the function which is very near to a. If these values tend to some definite unique number as x tends to a, then that obtained unique number is called the limit of f(x) at x = a. This formula is applied in places where we need to calculate the derivative of a function. It is a source of making approximations to the actual value of the quantity as of close as possible.
2. Who Invented Limits?
Archimedes' thesis, The Method, was lost until 1906 when mathematicians discovered that Archimedes came close to discovering infinitesimal calculus. As Archimedes' work was not known until the twentieth century, then others developed the modern mathematical concept of limits. The idea of limits was developed by Archimedes to measure the volume of spheres and curved figures. His idea was that if he carves these figures on a small piece and then increases the number of pieces, then, in that case, the limit of the sum of pieces will give the desired result.
3. What is the Limit of a Constant?
The limit of constant times a function is equal to the constant times the limit of the function. We know that the limit of a product is always equal to the product of the limits. The limit of a quotient is equal to the quotient of the limits. The limit of any given constant function is equal to the constant. For a limit to exist in an equation, there should be a left-hand limit and a right-hand limit equal to each other. The two limits should also be equal to the value of the function at that point. Therefore, using this definition, we can state that the limit of a constant is itself.
4. What is Infinity Minus Infinity?
It is very natural for you to think that infinity minus infinity is zero because any number when subtracted to itself results in zero. But infinity is not a real, that is, rational number and therefore, when it is subtracted to itself, it does not result in zero. It is impossible for infinity subtracted from infinity to be equal to one and zero. Using such type of math, we can get infinity minus infinity as equal to any real number. Therefore, infinity subtracted from infinity is undefined.
5. How can I find the limit of a function containing a polynomial?
Polynomial functions are often asked in exams for the students to calculate their limits. Calculating the limits of a polynomial function is an easy task if you follow the steps given below, that is
Break the polynomial expression into individual terms using the properties of limits.
Now, find the limits of each of the individual terms.
Now add the limits of each of the individual terms together to get the limit of the function containing a polynomial expression.
An alternative method to this is by evaluating the function for a.