AIEEE Concepts®: Limits & Derivatives

Limits & Derivatives

Limits

The concept of limit of a function is best understood if one can distinguish between the statements; "the value of f (x) at x = a" and "the value of f (x) as x

a

(i.e. x tends to a or x approaches a)". x

a (or x approaches a) implies that |x - a| <

, where

can be made as small as desired. The interval (a -

, a +

) is called the neighbourhood of a.

Let y = f(x) be a given function defined in the neighbourhood of x = a, but not necessarily at the point x = a. The limiting behaviour of the function in the

neighbourhood of x = a when |x - a| is small, is called the limit of the function when x approaches 'a' and we write this as

f(x).

Remarks:

(i)Often, f(x) can be simplified by using series expansion, rationalization or using conjugate surds, before limit is obtained.

(ii)For the existence of the limit at x = a, f(x) need not be defined at x = a. However if f(a) exists, limit need not exist or even if it exists then it need not be equal to f(a).

Algebra of limits

The following are some of the Basic Theorems on limits which are widely used to calculate the limit of the given functions.

Let

f(x) =

₁ and

g(x) =

₂ where

₁ and

₂ are finite numbers. Then

(c₁f(x)

c₂g(x)) = c₁

₁

c₂

₂, where c₁and c₂ are given constants.

f(x).g(x) =

f (x).

g (x) =

₁.

₂.

.

Note: If

exists, then it is not always true that

will exist.

Some Important Results on Limits

If p(x) is a polynomial,

p(x) = p(a).

, a > 0

cosx = 1 (where 'x' is in radians)

For two function within the same domain, we say that f < g if f(x) < g(x) for all x in the domain.

If both

exist, then f

and f, g, h are real functions such that f

h for all points in an open interval containing a, then

. (Sandwitch theorem)

If for all points in an open interval containing a, 0

g and

, then

if and only if

.

Limit at infinity

In case we want to find the limit of a function f (x) as x takes larger and larger values, we write

. If the degree of the denominator in a rational

(polynomial) function is higher than the degree of the numerator, then the

is zero. If the degree of the denominator is lower than the degree of the numerator,

then the

.

Left and right hand limits
If the limit of f (x) as x

a⁺, (indicating that x approaches a from values of x greater than a), exists, we write

= l₁, h > 0.

If the limit of f (x) as x

a^-, (indicating that x approaches a from values of x lesser than a), exists, then we write

(a - h) = l₂, h > 0.

The limits l₁ and l₂ are called the right-hand limit and the left-hand limit respectively. When l₁ = l₂ = l, then the function f (x) has a limit as x

a, and we write

f(x) = l.

If l₁

l₂, then the

f(x) does not exist.

Differentiability

The derivative or differential coefficient of f(x) w.r.t. x at x

(a, b), denoted by dy/dx or f '(x), is

... (1)

provided the limit exists and is finite; and the function is said to be differentiable.

Right Hand Derivative

Right hand derivative of f(x) at x = a is denoted by, Rf'(a) or f'(a⁺) and is defined as

R

, h > 0.

Left Hand Derivative

Left hand derivative of f(x) at x = a is denoted by

and is defined as

, h > 0.

Clearly, f(x) is differentiable at x = a if and only if R f '(a) = Lf '(a).

To find the derivative of f(x) from the first Principle

, dx > 0.

It implies that if the right-hand derivative and the left-hand derivative exist and are equal, the function f (x) is said to be differentiable at x = x₀ and has the derivative f' (x₀). When the derivative of a function is obtained directly by using the above definition of the derivative, then it is called differentiation from the first

principle.

List of derivatives of important functions: