Types of Limits
Limits in maths are of several types, each describing different situations and behaviors of functions as the independent variable approaches a certain value or infinity. Here are the main types of limits:
One-Sided Limits
There are two path to approach any point in 2D space along a curve. That are from Left Hand Side of Curve or Right Hand Side of Curve. Approaching the curve from either sides allow us to find two separate limit of the function. These two limits are called,
- Left Hand Limit
- Right Hand Limit
Now, let’s learn about them in detail.
LHL (Left Hand Limit)
The the left-hand limit of the function is defined as the vale of the function when the variable approaches the value from left side of the function. It is represented as,
lim x ⇢ a- f(x) = L
RHL (Right Hand Limit)
The the right-hand limit of the function is defined as the vale of the function when the variable approaches the value from right side of the function. It is represented as,
lim x ⇢ a+ f(x) = L
Two-Sided Limits
Two-sided limits, also known as bilateral limits, are a fundamental concept in calculus that describe the behavior of a function as the independent variable approaches a particular value from both the left and the right sides simultaneously.
Formally, let f(x) be a function defined on an open interval containing x=c, except possibly at x=c itself. The two-sided limit of f(x) as x approaches c, denoted as:
limx→c f(x)
exists if and only if both the left-hand limit (as x approaches c from the left) and the right-hand limit (as x approaches c from the right) exist and are equal.
Mathematically, the two-sided limit L of f(x) as x approaches c is defined as follows:
limx→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.
In simpler terms, this definition means that as x gets arbitrarily close to c, f(x) gets arbitrarily close to L.
The concept of two-sided limits is essential for understanding continuity, determining the existence of limits at a point, and evaluating derivatives in calculus. It helps analyze the behavior of functions near specific points and understand their overall characteristics and behavior in various mathematical contexts.
Limits in Calculus
Limits in maths are defined as the values approaching the output for the given input values of a function. Limits are used in calculus and mathematical analysis for finding the derivatives of the function. They are also used to define the continuity of the function.
The limit of any function is also used to find the integral of the function. The integral are of two types, Indefinite Integral, and Definite Integral, in definite integral we use the concept of upper limit and lower limit to find the answer to the definite integral. A function can reach a particular value from more than one path and the value of the function at that particular point is called the limit of the function at that point. Suppose we are given a function f(x) and when x approaches a the function approaches A this i represented using the limit as:
lim x ⇢ a f(x) = A
In this article, we will learn the introduction to limits, properties of limits, limits, and continuity, and others in detail.
Table of Content
- What Are Limits in Calculus?
- Limits Definition
- Formula of Limit
- Types of Limits
- Infinite Limits
- Limits at Infinity
- Properties of Limits
- Algebra of Limit
- Limits and Functions
- Limit of Polynomial Function
- Limit of Rational Function
- Limits of Complex Functions
- Limits of Exponential Functions
- Limit of a Function of Two Variables
- Calculating Limits
- Examples of Limits
- Limits and Derivatives Class 11
- Resources Related to Limits:
- Practice Questions on Limits in Calculus