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Table of contents
1.
Introduction
2.
Limits
2.1.
Types of Limit
2.2.
Existence of Limit
3.
Continuity
4.
Differentiability
5.
FAQs
6.
Key Takeaways
Last Updated: Mar 27, 2024

Limits, Continuity and Differentiability

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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=limx→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. 

  1. 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.
  2. 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).

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Continuity

A real-valued function f(x) is continuous at a point x=a only if limx→a f(x) exists and is equal to f(a). A continuous function's graph is always a single unbroken curve. For a continuous function f(x), the following holds-

limx→a- f(x) = limx→a+ f(x) = limx→a f(x). So, the basic difference between limits and continuity is that the function needs to be defined at that particular point to be continuous, and the limiting value should be equal to f(a). 

Any mathematical operation(sum, difference, product, or quotient) on two continuous functions is always continuous. But, in the case of the quotient of two functions, the denominator must be zero.

If a function is not continuous, it is said to be a discontinuous function. A function is discontinuous in any of the following cases:

1. lim x→a- f(x) and lim x→a+ f(x) exist but are not equal.

2. lim x→a- f(x) and lim x→a+ f(x) exist and are equal but not equal to f(a).

3. f(a) is not defined.

4. At least one of the limits does not exist. 

Differentiability

The derivative of a real-valued function f(x) with respect to x is the function f'(x) and is defined as limh→0 f(x+h)-f(x)/h. If the derivative of a function exists at all points of its domain, the function is said to be differentiable. Let us take a point x=c. Differentiability of a function f(x) can be determined only if limh→0 f(x+c)-f(x)/c exists. 

For a closed interval [a, b], a function f is said to be differentiable if for the points a and b, f'(a+) and f'(b-) exist and for any point c such that a<c<b, f'(c+) and f'(c-) exist and are equal. A differentiable function is always continuous, but a continuous function may not always be differentiable.

FAQs

  1. What is limit, continuity and differentiability?
    The limit of a function at a point x=a is the value that function achieves at the point closest to x=a. The function f(x) is continuous at the point x=a only if lim x→a f(x) exists and is equal to f(a).The derivative of the function f(x) with respect to x is the function f'(x) and is defined as lim h→0 f(x+h)-f(x)/h.
  2. How do you know if a limit is differentiable?
    A function is considered differentiable if its derivative exists at each point in its domain.

Key Takeaways

In this article, we have extensively discussed the concept of limits, continuity, and differentiability along with a few examples.

Recommended Readings:

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Happy Learning! 

 

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