rules of limits

8/6/2019 Rules of Limits

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Rules of LimitsWe have seen in the previous section that proving that the limit of

a function at some value of x has a specific value is really a tedious task. Moreover,

the method we used cannot find us the limit of the function, but only prove thevalue we know is true. In the following section, we will see how to find the limits

of most of the functions we know. Let¶s see the theorems we use for this object. Theore

m The limit of the constant function , where c is a constant, at any

value of x is equal to c. That is, Theore

m The limit of the identity function at some point is equal to thevalue of x at that point, or, in symbols:

Theore

m If we have two functions f and g whose limits exist at a number a, then, 1.

2.

3.

4. , in condition that n is any positive integer,

and when n is even. That is a very important theorem to calculate the limits of many functions.

It leads us also to the following corollaries. Corollar

y If exists and k is a constant, then: Corollar y If exists and n is any real number , then: Example Find:

Solution Therefore,

The followin g two corollaries ar e very helpful.

Corollary Let f be the polynomial function defined

by: , wheren is a positive integer

and are constants. Then the limit of the function f at any number c is:



Corollary Let f be a rational function, and c a number in its domain, then NB The limit of a function at some point has a unique value. No limit can have

two values at the same time. All what we saw before cannot yet be exploited except if the function is

defined at the point of the limit. What about the limit when the function has anindeterminate value at some point? The following theorem is the answer.

Theorem If we have two functions, f and g , such that , for all

except possibly at a, and exists, we can say that T

he use of this theorem is indicated by the following example.

Example Let f be the function defined by: Find:

Solution

Theorem

The function f is a rational function, but 1/3 is not an element of its domain.Hence, we cannot simply substitute in the previous rule of correspondence to get

the limit, but using the theorem we have just learnt will make the answer easy. First, note that the function can be rewritten in the form

Now, consider the function , for all .The two

functions f and g have the same value at all real numbers except 1/3. Thus, we can say that: Lim f(x) = lim g(x) = lim (x^2 + 4x ± 3) = 1/9 + 4/3 ± 3 = 14/9

Thus lim f(x) = g(x)

Let f be the function defined by: such that a is a constant, and n is any real number. Then,

Corollar

y If n , m and a are any real constants, then, Exampl

e Evaluate the limit: Solution



To calculate the limits of some trigonometric functions, the following

theorem and corollaries are needed.

Theore

m If x is an angle measured in radians, we say that 1.

2.

Corollar y If x is an angle measured in radians, and a and b are constants, then

1. . 2. . Exampl

e Calculate:

Limit Definition of the Derivative

Once we know the most basic differentiation formulas and rules, we compute newderivatives using what we already know. We rarely think back to where the basic

formulas and rules originated.

The geometric meaning of the derivative

f ( x)= df ( x) / dx

is the slope of the line tangent to y= f ( x) at x.

Let's look for this slope at P :

The secant line through P and Q has slope

f ( x + x ) f ( x ) = f ( x + x ) f ( x )

( x + x ) x x

We can approximate the tangent line through P by

moving Q towards P , decreasing x. In the limit as x 0, we get the tangent line

through P with slope

lim x 0 f ( x + x ) f ( x )

x We define

f¶ ( x)=lim x 0 f ( x + x ) f ( x )

x



If the limit as x 0 at a particular point does not exist, f ¶( x) is undefined at that

point.

We derive all the basic differentiation formulas using this definition.

Example

For f ( x)= x2,

f ( x) = lim x 0 (( x+ x)^2í x^ 2) / x

= lim x 0 (( x^ 2+2( x) x+ x^ 2)í x^ 2) / x

= lim x 0 2( x) x+ x^ 2 / x

= lim x 0(2 x+ x)

= 2 x

as expected.

Example

For f ( x)= x1

f ( x) = = = = = lim x 0 x1 x+ xí x1 lim x 0 x xí( x+ x)( x+ x)( x) lim x 0 xí

x( x+ x)( x) lim x 0í1( x+ x)( x) í1 x2

again as expected.

Notes

The limit definition of the derivative is used to prove many well-known results,

including the following:

y If f is differentiable at x0, then f is continuous at x0.

y Differentiation of polynomials: d / dx x^ n =n x^ ní1 .y Product and Quotient Rules for differentiation.

K ey Concepts

We define f ( x)=lim x 0 f ( x+ x)í f ( x).

x

rules of limits

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