Limits
In Mathematics, a limit is defined as a value that a function approaches the output for the given input values. Limits are important in calculus and mathematical analysis and used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns the behavior of the function at a particular point. The limit of a sequence is further generalized in the concept of the limit of a topological net and related to the limit and direct limit in the theory category.
Generally, the integrals are classified into two types namely, definite and indefinite integrals. For definite integrals, the upper limit and lower limits are defined properly. Whereas in indefinite the integrals are expressed without limits, and it will have an arbitrary constant while integrating the function. In this article, we are going to discuss the definition and representation of limits, with properties and examples in detail.
Limits Definition
Let us consider a realvalued function “f” and the real number “c”, the limit is normally defined as \(\lim _{x \rightarrow c} f(x)=L\). It is read as “the limit of f of x, as x approaches c equals L”. The “lim” shows the limit, and fact that function f(x) approaches the limit L as x approaches c is described by the right arrow.
Limits and Functions
A function may approach two different limits. One where the variable approaches its limit through values larger than the limit and the other where the variable approaches its limit through values smaller than the limit. In such a case, the limit is not defined but the right and lefthand limit exists.
 The righthand limit of a function is the value of the function approaches when the variable approaches its limit from the right. This can be written as: \[\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\mathrm{A}^{+}\]
 The lefthand limit of a function is the value of the function that approaches when the variable approaches its limit from the left. This can be written as: \[\lim _{x \rightarrow a} \mathrm{f}(\mathrm{x})=\mathrm{A}^{}\]
 The limit of a function exists if and only if the lefthand limit is equal to the righthand limit. \[\lim _{x \rightarrow a^{1}} \mathrm{f}(\mathrm{x})=\lim _{x \rightarrow a^{+}} \mathrm{f}(\mathrm{x})=\mathrm{L}\]
Note: The limit of this function exists between any two consecutive integers.
Properties of Limits
Here are some properties of the limits:
\(\begin{array}{l}
\text { If limits } \lim _{x \rightarrow a} \mathrm{f}(\mathrm{x}) \text { and } \lim _{x \rightarrow a} \mathrm{~g}(\mathrm{x}) \text { exists, then, }\\
\begin{array}{ll}
\hline \text { Law of Addition } & \lim _{x \rightarrow a}[f(x)+g(x)]=\lim _{x \rightarrow a} f(x)+\lim _{x \rightarrow a} g(x) \\
\hline \text { Law of Subtraction } & \lim _{x \rightarrow a}[f(x)g(x)]=\lim _{x \rightarrow a} f(x)\lim _{x \rightarrow a} g(x) \\
\hline \text { Law of Multiplication } & \lim _{x \rightarrow a}[f(x) \cdot g(x)]=\lim _{x \rightarrow a} f(x) \cdot \lim _{x \rightarrow a} g(x) \\
\hline \text { Law of Division } & \lim _{x \rightarrow a}\left[\frac{f(x)}{g(x)}\right]=\frac{\lim _{x \rightarrow a} f(x)}{\lim _{x \rightarrow a} g(x)}, \text { where } \lim _{x \rightarrow a} g(x) \neq 0 \\
\hline \text { Law of Power } & \lim _{x \rightarrow a} c=c \\
\hline
\end{array}
\end{array}\)
Where, n is an integer.
Special Rules:
1. \(\lim _{x \rightarrow a} \frac{x^{n}a^{n}}{xa}=n a^{(n1)}$, for all real values of $\mathrm{n}\).
2. \(\lim _{\theta \rightarrow 0} \frac{\sin \theta}{\theta}=1\)
3. \(\lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta}=1\)
4. \(\lim _{\theta \rightarrow 0} \frac{1\cos \theta}{\theta}=0\)
5. \(\lim _{\theta \rightarrow 0} \cos \theta=1\)
6. \(\lim _{x \rightarrow 0} e^{x}=1\)
7. \(\lim _{x \rightarrow 0} \frac{e^{x}1}{x}=1\)
8. \(\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e\)
Limit of a Function of Two Variables
If we have a function f(x, y) which depends on two variables x and y, then this given function has the limit, say, C as (x,y) → (a,b) provided that \(\epsilon\) > 0, there exits \(delta\) > 0 such that f(x, y)C < \(\epsilon\) whenever 0<\(\sqrt{(xa)^{2}+(yb)^{2}} < \delta\) . It defined as \(\lim _{(x, y) \rightarrow(a, b)}\) f(x,y) = C
Limits of Functions and Continuity
Limits and continuity are closely related to each other. Functions can be continuous or discontinuous. For a function to be continuous, if there are small changes in the input of the function then must be small changes in the output.
In elementary calculus, the condition \(f(X)\lambda \lambda\) as \(x \rightarrow a\) means that the number \(f(x)\) can be made to lie as close
as we like to the number lambda as long we take the number $x$ unequal to the number a but close enough to a. Which shows that \(\mathrm{f}(\mathrm{a})\) might be very far from lambda and there is no need for \(\mathrm{f}(\mathrm{a})\) even to be defined. The very important result we use for the derivation of function is: \(\mathrm{f}^{\prime}(\mathrm{a})\) of a given function \(\mathrm{f}\) at a number a can be thought of as,
\(\mathbf{f}^{\prime}(\mathrm{a})=\lim _{x \rightarrow a} \frac{f(x)f(a)}{xa}\)
Limits of Complex Functions
To differentiate functions of a complex variable follow the below formula:
The function \(f(z)\) is said to be differentiable at \(z=z_{0}\) if
\(\lim _{\Delta z \rightarrow 0} \frac{f\left(z_{0}+\Delta z\right)f\left(z_{0}\right)}{\Delta z}\) exists. Here \(\Delta \mathrm{z}=\Delta \mathrm{x}+\mathrm{i} \Delta \mathrm{y}\)
Limits of Exponential Functions
For any real number \(\mathrm{x}\), the exponential function \(\mathrm{f}\) with the base \(\mathrm{a}\) is \(\mathrm{f}(\mathrm{x})=\mathrm{a}^{\wedge} \mathrm{x}\) where \(\mathrm{a}>0\) and a not equal to zero. Below are some of the important limits laws used while dealing with limits of exponential functions.
For \(\mathbf{b}>1\)
 \(\lim _{x \rightarrow \infty} b^{x}=\infty\)
 \(\lim _{x \rightarrow\infty} b^{x}=0\)
For \(0<b<1\)
 \(\lim _{x \rightarrow \infty} b^{x}=0\)
 \(\lim _{x \rightarrow\infty} b^{x}=\infty\)
Related Topics:
Examples on Limits

Example 1: Check for the limit, \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\)
Solution:
Since we have modulus function in the numerator, so let us evaluate right hand and left hand limits first.\(\mathrm{RHL}=\lim _{h \rightarrow 0+} \frac{\sin (h)}{h}=1\)
\(\mathrm{LHL}=\lim _{h \rightarrow 0^{}} \frac{\sin (h)}{h}=1\)
As RHL and LHL revert different values, therefore, the given limit does not exist.

Example 2: Find the limit. lim_(x→0) (tan x)/(sin x)
Solution: lim_(x→0) (tan x)/(sin x)
We know, tan x = sin x / cos x
= lim_(x→0) (1/cos x)
= 1
=> lim_(x→0) (tan x)/(sin x) = 1

Example 3: Evaluate lim_(x→0) sin(2x)
Solution:
lim_(x→0) sin(2x) = lim_(x→0) [sin(2x)/2×(2)]
= 2 lim_(x→0) sin(2x)/2
= 2
FAQs on Limits
What Is the Limit Formula?
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 a unique number is called the limit of f(x) at x = a.
What Are Limits in Calculus?
A limit tells us the value that a function approaches as that function's inputs get closer and closer(approaches) to some number. The idea of a limit is the basis of all calculus.
When Can a Limit Not Exist?
A common situation where the limit of a function does not exist is when the onesided limits exist and are not equal: the function "jumps" at the point. The limit at x→0 does not exist.
Why Do We Use Limits in Mathematics?
Limit, a mathematical concept based on the idea of closeness, used primarily to assign values to certain functions at points where no values are defined, in such a way as to be consistent with nearby values.
How Do You Know if a Limit Is OneSided?
A onesided limit is a value the function approaches as the xvalues approach the limit from *one side only*. For example, f(x)=x/x returns 1 for negative numbers, 1 for positive numbers, and isn't defined for 0. The onesided *right* limit of f at x=0 is 1, and the onesided *left* limit at x=0 is 1.
How Are Calculus Limits Used in Real Life?
Limits are also used as reallife approximations to calculating derivatives. So, to make calculations, engineers will approximate a function using small differences in the function and then try and calculate the derivative of the function by having smaller and smaller spacing in the function sample intervals.
What Is the Limit of a Sine Function?
Since sin(x) is always somewhere in the range of 1 and 1, we can set g(x) equal to 1/x and h(x) equal to 1/x. We know that the limit of both 1/x and 1/x as x approaches either positive or negative infinity is zero, therefore the limit of sin(x)/x as x approaches either positive or negative infinity is zero.