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Limits and 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 x alim 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 x alim f(x) = 1 and

x alim g(x) = 2 where 1 and 2 are finite numbers. Then

x alim (c1f(x) c2g(x)) = c1 1 c2 2, where c1and c2 are given constants.

x alim f(x).g(x) =

x alim f (x).

x alim g (x) = 1. 2.

x alim

x a 1

22

x a

lim f(x)f(x)

, 0g(x) lim g(x)

.

Note: If x alim f(x) g(x) exists, then it is not always true that

x a x alim f(x) lim g(x) and

will exist. Some Important Results on Limits

If p(x) is a polynomial, ax

lim p(x) = p(a).

n nn 1

x a

x alim na

x a, a > 0

n

x 0

(1 x) 1lim n

x.

m m m mm 1

n n n n n 1x a x a

x a x a x a 1lim lim ma

x ax a x a na = m nm

an

.

2

0 0 0

sinx tan x

x xx x xlim lim lim cosx = 1 (where „x‟ is in radians)

x 0 x 0 x 0 x 0

tanx sinx 1 sinx 1lim lim lim lim 1

x x cosx x cosx

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 x a x alim f x and lim g x exist, then f g

x a x alim f x lim g x .

If x a x alim f x l lim h x and f, g, h are real functions such that f g h for

all points in an open interval containing a, then x alim g x l . (Sandwitch

theorem)

If for all points in an open interval containing a, 0 f g and x alim g x 0 ,

then x alim f x 0 .

x alim f x 0 if and only if

x alim f x 0 .

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 xlim f(x) . If the degree of the denominator in a rational (polynomial) function

is higher than the degree of the numerator, then the xlim is zero. If the degree of the

denominator is lower than the degree of the numerator, then the xlim .

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 h 0x a

lim f(x) lim (a h) = 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

h 0x a

lim f(x) lim (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 x alim f(x) = l.

If l1 l2, then the x alim 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

x 0

dy f(x x) f(x)

dx xlim ... (1)

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

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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 f a = h 0lim

f a h f a

h, h > 0.

Left Hand Derivative

Left hand derivative of f(x) at x = a is denoted by Lf a or f a and is defined as

Lf a = h 0lim

f a h f a

h, 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

0

0 0 0 0

x 0 x 0x x

f(x x) f(x ) f(x x) f(x )dylim lim

dx x x, x > 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:

n n 1d

(x ) nxdx

, n R

d

(sinx) cosxdx

d

(cosx) sinxdx

2d

(tanx) sec xdx

2d

(cot x) cosec xdx

d

(sec x) sec x tanxdx

d

(cosecx) cosecx cot xdx

Product of two functions f(x) and g(x). If y = u (x) v (x) w (x), then

dy d dw

(u v) w(x) uvdx dx dx

= uw dv du dw

vw uvdx dx dx

If g (x) = f (x), then y = (f (x))2 dy df

2f(x)dx dx

If u (x) = v (x) = w (x) = f (x), then y = [f (x)]3

4

and 2 2 2 2dy df df df df

f(x) f(x) f(x) 3 f(x)dx dx dx dx dx

.

In general, for y = (f (x))n, n R,

n 1dy dfn(f(x))

dx dx