Limits, Continuity and Differentiability

Limits, Continuity and Differentiability

Introduction

This topic educates the learners on limits, continuity, and differentiability, the existence of limits, expansion in evaluating limits, and evaluation of algebraic limits. This guide will also help you understand the basics of function graphs.

Limits, continuity, and differentiability are integral and essential parts of mathematics. 

Let us start by defining the limits. 

What are the Limits?

A limit can be described as the value when a function approaches a given input value. They are important in mathematical analysis and calculus. They are used in describing derivatives, continuity, and integrals. xcf(x) = L is an expression that is used for describing the limits xcf(x) = L , where the limit of f(x) as x approaches c equals L. The “lim” represents the limit, and the fact that function f(x) approaches the limit L as x approaches c is shown by the right arrow.

What are the properties of limits?

Now let us discuss some of the essential properties of limits:

It can be assumed that  xdf(x) and xdg(x)  exist, and d is a constant. Then, 

1. xd[c.f(x)] = cxdf(x) 

We can factor a constant which is multiplicative out of the limit.

  2. xd[f(x) ± g(x)] =  xdf(x) xdg(x)

To consider the limit of a difference or sum, we can choose the limits individually and put them back with the corresponding sign. This helps regardless of the volume of functions we separate by “+” or “-”.

  3. xd[f(x). g(x)] =  xdf(x).xdg(x)

Consider the limits of products same as the limits of sums or differences. Just select the limit of the individual functions and put them back. This is not limited to only two functions.

  4. xd[f(x)g(x)] =  xdf(x)xdg(x)

 Provided xdg(x) ≠ 0

As can be seen here, if the limit of the denominator is not zero when operating the quotient limit, we can separate the limits. If it is zero, it results in a division by zero error.

   5. xdc = c

The limit of a constant variable is a constant. You can easily understand it by plotting a graph of the function f(x) = c.

   6. xd xn= dn

Existence of limits

The existence of limits will depend on whether the following conditions are fulfilled:

1. Both the left-hand side and the right-hand side should be finite.
2. xd-f(x) = xd+f(x) i.e. LHS =RHS
3. In Limits, we have indeterminate forms such as   etc.

In these conditions, we try to simplify the question into a proper function.

Expansion in Evaluating Limits

Some of the essential limits are as follows:

1. x0sin xx = 1
2. x0tan xx = 1
3.  x01 -cos xx2 =12
4. x0ex – 1x = 1
5. x0log(1 + x)x = 1

Some of the critical expansions are as follows:

1. Log (1 + x) = x – x22 + x33 – x44 + …..
2. ex = 1+ x22! + x33! – x44! + …
3. ax = 1+ x log a + x22!(log a)2 + ……
4. Sin x = x –   x33! +  x55!……
5. Cos x = 1- x22! + x44! ……
6. Tan x = x + x33 + 215 x5 + ….

Evaluation of Algebraic Limits

1. Direct Substitution Method 
2. Rationalisation Method
3. Factorisation method 

What is continuity?

Qualitatively the graph of a function is said to be continuous at x =  d if travelling along with the graph of the function and crossing over the point at x =  d either from left to right or from right to left, and one does not have to lift his pen. If one has to lift his pen, the function’s graph has a break or discontinuous at x = d.

A function f(x) is continuous at 

x = d, if

xd-f(x) = xd+f(x)= f(d)

i.e. Left-Hand limit = Right-Hand Limit = value of the function at x = d

Otherwise, the function f(x) is a discontinuous function. The theory of limit and continuity is one of the essential terms to understand calculus. A limit is stated as a number that a function achieves as the independent variable of the function approaches a given value.

What is Differentiability?

A function f(x) is explained to be differentiable at the point x = d if the derivative f’(d) exists at every point in its domain.

The function f(x) is said to be non-differentiable at x = d if

(a) Both R.H.D & L.H.D exist but are not equal

(b) Either or both R.H.D & L.H.D are not limited

(c) Either or both L.H.D & R.H.D do not exist.

Conclusions

Limits, continuity, and differentiability are essential terms in mathematics to do calculus. They are interrelated terms, i.e., if a function is continuous, we can only talk about differentiability. And limits define the term continuity. Here, we learned about what are limits, properties of limits, Existence of limits, Expansion in Evaluating Limits, some of the essential expansions, Evaluation of Algebraic limits, continuity, and differentiability