## Introduction

The concept of limits has been discovered for thousands of years for a better understanding of calculus. The idea of limits arose from two critical problems: the first being the tangent problem, i.e., determining the slope of a line tangent to a curve at a point, and the second being the area problem, i.e., determining the area under a curve. But, until the late 19th century, the formal definition of limits was not defined.

## Limits

Limits mean "to set bounds for." limit implies setting a point or line in terms of time, space, speed, or degree beyond which something is not permitted. In Mathematics, the limit of a function at a point x=a is the value that function achieves at the point closest to x=a. Limits are crucial in calculus and mathematical analysis and definite integrals, derivatives, and continuity.

Let f(x) be a function defined on an interval containing the point p, except that it may not be determined at the point p. Let us assume L as a real number, and the function f is said to tend to a limit L written as L=lim_{x→p} f(x) if there is a number δ for every number ∈ such that |f(x)-L| < ∈ whenever 0 < |x-a| < δ.

### Types of Limit

Limits can be approached from either side of a number. Therefore, it can be defined in terms of a number either less than or greater than the given point. Based on these criteria, there are two different types of limits. Suppose the given point is x=a.

**Left-Hand Limit –**If the limit is defined in terms of a number less than the given number,i.e., a, then the limit is said to be the left-hand limit. It is denoted as x→a-, equivalent to x=a-h where h>0 and h→0.**Right-Hand Limit –**If the limit is defined in terms of a number greater than the given number,i.e., a, then the limit is said to be the right-hand limit. It is denoted as x→a+, equivalent to x=a+h where h>0 and h→0.

### Existence of Limit

The limit of a function f(x) at x=a exists only if its left-hand limit and right-hand limit exist and are equal having a finite value, i.e.,

lim x→a^{-} f(x) = lim x→a^{+} f(x).