Limit and Continuity

The concept of limit and continuity gave birth to calculus. Without limits,

calculus would not exist. Every single notion of calculus is a limit in one sense or

another. All different branches of calculus have evolved from the idea of limit.

Continuity can’t be defined without the concept of limits. “Limit and continuity” is one of the most

important chapter of the mathematics syllabus of JEE Advanced, JEE Main and other engineering

entrance exams. These concepts are not only used in the calculus but also in various other fields of

science like physics, chemistry, computer science etc. This chapter is the basic and easiest portion of

calculus and it can fetch you excellent scores. It is very important for a JEE aspirant to master the

concepts of this chapter if he/she wants to score good marks in mathematics of JEE. Beginners are

advised to refer the study material on limit and continuity.

Important Concepts

The Idea of a Limit

We start with a function f defined in the neighborhood of a number ‘c’ but not necessarily at ‘c’. Now let

L be some real number. We say that the limit of f(x) as x tends to c is L and write

If as x approaches c, f(x) approaches L.

Graphical Interpretation

We have already witnessed that in taking the limit of f(x) as x tends to c, it doesn’t matter whether f is

defined at ‘c’ or not. The only thing that matters is the value attained by f at c. There are three different

cases shown in the figure given below. In the first case, f (c) = L. In the second case, f is not defined at c.

In the third case, f is defined at c, but f (c) ≠ L. However, in each case

Because, as suggested by the figures, as x approaches c, f (x) approaches L.

limx c

f x L

limx c

f x L

One – sided Limit

While dealing with limits, the numbers x tending to c can broadly be divided into two categories: those

which lie to the left of c and those which lie to the right of c. We write

to indicate that as x approaches c from the left, f(x) approaches L. We write

to indicate that as x approaches c from the right, f(x) approaches L.

Algebra of Limits

limx→a

k = k

limx→a

kf(x) = k limx→a

f(x)

limx→a

(f(x) ± g(x)) = limx→a

f(x) ± limx→a

g(x)

limx→a

f(x). g(x) = limx→a

f(x) limx→a

g(x)

limx→a

f(x)

g(x)=

limx→a

f(x)

limx→a

g(x)

limx→a

bf(x) = blimx→a

f(x) [provided it is not of 1∞ form]

limx→a

f(g(x)) = limx→b

f(x) where b = limx→a

g(x)

Some Important Limits:

lim𝑥→𝑎

(𝑥𝑛−𝑎𝑛

𝑥−𝑎) = 𝑛𝑎𝑛−1

lim𝑥→0

𝑒𝑥−1

𝑥= 1

lim𝑥→0

𝑎𝑥−1

𝑥= ln 𝑎

lim𝑥→0

ln (1+𝑥)

𝑥= 1

lim𝑥→0

𝑙𝑜𝑔𝑎(1+𝑥)

𝑥= 𝑙𝑜𝑔𝑎𝑒

limx c

f x L

limx c

f x L

L = lim x→a

f′(x)

g′(x)

If h(x) < 𝑔(x) < 𝑓(𝑥)

and limx→a

h(x) limx→a

f(x) = L

then limx→a

g(x) = L

lim𝑥→0

𝑠𝑖𝑛𝑥

𝑥= 1

lim𝑥→0

𝑡𝑎𝑛𝑥

𝑥= 1

lim𝑥→0

𝑐𝑜𝑠𝑥 = 1

lim𝑥→0

𝑠𝑖𝑛−1𝑥

𝑥= 1

lim𝑥→0

𝑡𝑎𝑛−1𝑥

𝑥= 1

lim𝑥→0

(1 + 𝑥)1

𝑥 = 𝑒

lim𝑥→∞

(1 +1

𝑥)

𝑥

= 𝑒

L’Hospital’s Rule:

L = limx→a

f(x)

g(x) L = lim

x→a

f(x)

g(x)

If f(x)

g(x) is of 0/0 from, then

Squeeze / Sandwich Theorem

Continuity:

A function f is continuous at the point x = a if the following are true:

f(a) is defined

limx→a

f(x) exists

limx→a

f(x) = f(a)

limx→a−

f(x) = limx→a+

f(x) = f(a)

Nature of Discontinuity

The following chart illustrates the various types of discontinuities:

The discontinuity in Fig. shown below is called jump discontinuity.

The following figures illustrate functions exhibiting infinite discontinuities.

Theorems

If f and g are continuous at c, then

f + g is continuous at c;

f – g is continuous at c;

af is continuous at c for each real a

f.g is continuous at c

f/g continuous at c provided g(c) ≠ 0

Nature of Discontinuity

Discontinuity of First Kind

Discontinuity of Second

Kind

Removable Discontinuity

Jump Discontinuity

Non-Removable

Discontinuity

One sided Continuity

A function f called continuous from left at c if

A function f called continuous from right at c if

Continuity on Intervals

A function f is said to be continuous on open interval (a, b) if it is continuous at each and every

points in (a, b).

A function f is said to be continuous on closed interval [a, b] if it is

Right continuous at ‘a’

Left continuous at ‘b’

Continuous on (a, b)

Some Interesting Facts

If the limit of a function exists at some point, it doesn’t necessarily mean that the function

is continuous at that point.

If the function is continuous at some point, the limit of the function will definitely exist at

that point.

A function is considered continuous if it is continuous at all points of its domain.

Polynomial functions are always continuous.

Rational functions are continuous if the denominator is not equal to zero.

The sum, difference, product and quotient (as long as denominator is non-zero) of two

continuous functions is continuous.

lim𝑥→𝑐−

𝑓(𝑥) = 𝑓(𝑐)

lim𝑥→𝑐+

𝑓(𝑥) = 𝑓(𝑐)

If f is continuous on the closed interval [a, b], then f attains every value between f(a) and

f(b) at least once on the open interval (a, b).

We have various types of indeterminable forms in limit: 0

0,

∞

∞, 0 × ∞, ∞ − ∞, 00, 1∞, ∞0

Limit of sum is the sum of limits.

A function is continuous if you draw it without lifting your pen from the paper.

Integral part of x ([x]) has jump discontinuity at every integral point.

When the left hand limit and right hand limit at the point are not equal, then we have

‘finite irremovable discontinuity’ at the point.

‘Removable discontinuity’ situation comes when the limit exists at the point but

somehow the function is either not defined or defined wrongly at the point.

If the function is continuous at some point, it does not necessarily mean that it’s going to

be differentiable at that point.

Is limit and continuity an important part of IIT – JEE preparation?

‘Limit and continuity’ is very important and a pre-requisite for many topics of calculus of JEE

mathematics. Without the proper understanding of this chapter, students can’t perform well in

JEE mathematics as it accounts for around 15 – 20% weightage of calculus in JEE mathematics.

What are the best books for the preparation of limit and continuity?

Some of the books which are considered to be best for preparation of this section are:

I.A. Maron

Arihant

Tata McGraw Hill (TMH)

Tips to study Limit and Continuity for IIT JEE

This is the area of mathematics where students often make lots of silly

mistakes. You really need to master your basic concepts in this chapter if you

want to excel in calculus. These are some important and necessary tips you must follow to score

high in this area of JEE mathematics.

Practice as many problems as possible using different approaches like standard

expressions, L’Hospital rule, series expansion etc.

The easiest way to analyze a problem in limits is by using the series expansion of the

function.

Always go by basic root in the problems involving limits of integral part functions.

Important results and series expansions should be on your fingertips.

Always calculate the one – sided limit and continuity using the basic definition.

Do not refer too many books as it will lead to confusion. Students are advised to stick to

just one or two standard books.

Try to apply L’Hospital’s rule in solving problems as it saves time.

Try to attempt maximum number of problems from the reference books before solving

the previous year IIT – JEE questions.

Practical Applications of Limit and Continuity

Calculus is one of the most important branches of mathematics which has

various applications in the real world. Here are some of the examples:

The temperature at various instances during a day is a real life example of the continuous

function.

Calculus is used in modeling engineering problems which are often stated as ordinary or

partial differential equations. The solutions to these equations can be whole families of

curves, from which the engineer must select a physically meaningful particular solution.

A continuity condition is often imposed as one of the possible boundary conditions to

select the physically correct solution.

The concept of limits is used in the measurement of instantaneous speed of the vehicles

like motorcycle, car, truck etc.

A real life example of limit would be earth’s capacity to maintain the population of all

species. We consider the earth’s resources, the number of people and how much they

consume in comparison to what is available. As the limit is approached, or with the end

of resources, it eventually reaches zero and the population which needs it, dies.

Limit is used to calculate the area of the complex geometrical shapes where the basic

geometry fails in engineering.

The concept of limit and continuity is used for applications in the fields of astronomy and

time travel.

Engineers use calculus for building skyscrapers, bridges, robotics etc.

It is used to study the interactions of different species within the ecosystem.

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