Limits and Derivatives

Concept of a Function

y is a function of x, and the relation y = x2 describes a function. We notice that with such a relation, every value of x corresponds to one (and only one) value of y.

y = x2

Since the value of y depends on a given value of x, we call y the dependent variable and x the independent variable and of the function y = x2.

Notation for a Function : f(x)

The Idea of Limits

Consider the function

The Idea of Limits

2

4)(

2

x

xxf

x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1

f(x)

Consider the function

The Idea of Limits

2

4)(

2

x

xxf

x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1

f(x) 3.9 3.99 3.999 3.9999 un-defined

4.0001 4.001 4.01 4.1

Consider the function

The Idea of Limits 2)( xxg

x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1

g(x) 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1

2)( xxg

x

y

O

2

If a function f(x) is a continuous at x0,

then . )()(lim 00

xfxfxx

4)(lim2

xfx

4)(lim2

xgx

approaches to, but not equal to

Consider the function

The Idea of Limits

x

xxh )(

x -4 -3 -2 -1 0 1 2 3 4

g(x)

Consider the function

The Idea of Limits

x

xxh )(

x -4 -3 -2 -1 0 1 2 3 4

h(x) -1 -1 -1 -1 un-defined

1 2 3 4

1)(lim0

xhx

1)(lim0

xhx

)(lim0

xhx does not

exist.

A function f(x) has limit l at x0 if f(x) can be made as close to l as we please by taking x sufficiently close to (but not equal to) x0. We write

lxfxx

)(lim0

Theorems On Limits

Theorems On Limits

Theorems On Limits

Theorems On Limits

Exercise 12.1P.7

Limits at Infinity

Limits at Infinity

Consider1

1)(

2

xxf

Generalized, if

)(lim xfx

then

0)(

lim xf

kx

Theorems of Limits at Infinity

Theorems of Limits at Infinity

Theorems of Limits at Infinity

Theorems of Limits at Infinity

Exercise 12.2P.13

Theorem

where θ is measured in radians.

All angles in calculus are measured in radians.

1sin

lim0

Exercise 12.3P.16

The Slope of the Tangent to a Curve

The Slope of the Tangent to a Curve

The slope of the tangent to a curve y = f(x) with respect to x is defined as

provided that the limit exists.

x

xfxxf

x

yAT

xx

)()(limlim of Slope

00

Exercise 12.4P.18

Increments

The increment △x of a variable is the change in x from a fixed value x = x0 to another value x = x1.

For any function y = f(x), if the variable x is given an increment △x from x = x0, then the value of y would change to f(x0 + △x) accordingly. Hence thee is a corresponding increment of y(△y) such that △y = f(x0 + △x) –

f(x0).

Derivatives(A) Definition of Derivative.

The derivative of a function y = f(x) with respect to x is defined as

provided that the limit exists.

x

xfxxf

x

yxx

)()(limlim

00

The derivative of a function y = f(x) with respect to x is usually denoted by

,dx

dy),(xf

dx

d ,'y ).(' xf

The process of finding the derivative of a function is called differentiation. A function y = f(x) is said to be differentiable with respect to x at x = x0 if the derivative of the function with respect to x exists at x = x0.

The value of the derivative of y = f(x) with respect to x at x = x0 is denoted

by or .0xxdx

dy

)(' 0xf

To obtain the derivative of a function by its definition is called differentiation of the function from first principles.

Exercise 12.5P.21