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Limits (Intuitive Introduction)

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Calculus and Analytical Geometry

MTH101

LECTUER 9

Limits (Intuitive Introduction)

The line occupying this limiting position

we consider to be tangent line at P.

THE AREA PROBLEM

The areas of some plane regions can

be calculated by subdividing them into

Given a function f, find the area between

finite number of rectangles or triangle,

the graph of the f and an interval [a,b] on the

then adding the area of the constituent

x-axis

parts.

A line is called tangent

For many regions a more general approach

to a circle if

is needed.

it meets

the circle

at precise

ly one

point.

But this

definition

is not satis-

factory for

We approximate the area of this region by

other kind

inscribing rectangles of equal width under

of curves, like

The curve and adding the areas of these

rectangles.

The line is

tangent yet it

meets

the curve

more

than once.

Mth101

Page 24

Calculus and Analytical Geometry

Limits (Intuitive Introduction)

Our approximations will "approach" the

exact area under the curve as a "limiting

value".

As x approach 0 from the left

or right, f(x) approach 1.

Let us take

The preceding ideas are summarized in this table.

Numerical Pitfalls

Table 2.4.4

Using a calculator set to the radian mode, we have

Mth101

Page 25

Calculus and Analytical Geometry

Limits (Intuitive Introduction)

Existence of limits

Example

Here

x approaches X0 from left

So, limit of function does not exist.

Example

Here

x approaches X0 from right

So, limit of function does not exist.

Example

Example

Here

The limit of the function does not exist at X0

So, limit of function does not exist.

Mth101

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