Difference between revisions of "Applications-of-GeoGebra/C3/Limits-and-Continuity-of-Functions/English"

Title Slide

Learning Objectives

Understand limits of functions

Look at continuity of functions

System Requirement

Ubuntu Linux OS version 16.04

GeoGebra 5.0.481.0-d

Pre-requisites

www.spoken-tutorial.org

GeoGebra interface

Limits

Elementary calculus

For relevant tutorials, please visit our website.

Limits

[[Image:]][[Image:]]

Imagine yourself sliding along the curve or line towards a given value of x.

The height at which you will be, is the corresponding y value of the function.

Any value of x can be approached from two sides.

The left side gives the left hand limit.

The right side gives the right hand limit.

Limit of a rational polynomial function

Let us find lim (3x² – x -10)

x→2 (x² – 4)

Let us find the limit of this rational polynomial function as x tends to 2.

Let us graph functions and look at their limits.

Type (3 x^2-x-10)/(x^2-4) in the input bar >> Enter

Note that spaces denote multiplication.

In the input bar, first type the numerator.

Now, type the denominator.

Press Enter.

Click in and drag Graphics view to see the graph.

Click in and drag Graphics view to see the graph.

Remember to press Enter to go to the next cell.

Type 1.91, 1.93, 1.96, 1.98 and 2 in column A from cells 1 to 5.

We will choose values of x less than but close to 2.

Remember to press Enter to go to the next cell.

In column A in cells 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2.

Type 2.01, 2.03, 2.05, 2.07 and 2.09 in column A from cells 6 to 10.

We will choose values of x greater than but close to 2.

In column A from cells 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09.

First, the numerator in parentheses

3 A1 in parentheses caret 2 minus A1 minus 10 followed by division slash

Now the denominator in parentheses

A1 in parentheses caret 2 minus 4 and press Enter.

Place the cursor at the bottom right corner of the cell.

Drag the cursor to highlight cells until B10.

Point to y values in column B and to the x values in column A.

Place the cursor at the bottom right corner of the cell.

Drag the cursor to highlight cells until B10.

This fills in y values corresponding to the x values in column A.

Point to the spreadsheet.

This is because the function is undefined at this value.

The reason for this is that the denominator of the function becomes 0.

Observe that as x tends to 2, y tends to 2.75.

Hence, as x tends to 2, the limit of the function tends to 2.75.

Limits of discontinuous functions

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lim h(x) = ?

x→c

lim h(x) = L4; lim h(x) = L3

x→c- x→c+ Thus, lim h(x) Does Not Exist (DNE)

          x→c

In graph B, h of x is a piecewise or discontinuous function.

We want to find the limit of h of x as x approaches c.

So let us look at the left and right hand limits.

For the left hand limit, look at the lower limb where the limit is L4.

For the right hand limit, look at the upper limb where limit of h of x is L3.

But as x approaches c, the two limbs of h of x approach different values of y.

These are L3 and L4.

The left and right hand limits exist.

But the limit of h of x as x approaches c, does not exist (DNE).

Limit of a discontinuous function

Let us find lim f(x) = 2x+3, x ≤ 0

                x→0    3(x+1), x > 0

and lim f(x) = 2x+3, x ≤ 0

    x→1  3(x+1), x > 0

Let us find limits of a piecewise or discontinuous function f of x.

f of x is described by 2x plus 3 when x is 0 or less than 0.

But f of x is described by 3 times x plus 1 when x is greater than 0.

We want to find the limits when x tends to 0 and 1.

a equals Function with capital F and in square brackets 2x plus 3 comma minus 5 comma 0

This chooses the domain of x from minus 5 (for practical purposes) to 0.

Press Enter.

Drag the boundary to see it properly.

Point to its graph in Graphics view.

Drag the boundary to see it properly.

Its graph is seen in Graphics view.

When an arrow appears along the axis, drag the x-axis to zoom in or out.

When an arrow appears along the axis, drag the x-axis to zoom in or out.

When an arrow appears along the axis, drag the y-axis to zoom in or out.

When an arrow appears along the axis, drag the y-axis to zoom in or out.

Remember the space denotes multiplication.

b equals Function with capital F

In square brackets, type 3 space x plus 1 in parentheses comma 0.01 comma 5

This chooses the domain of x from 5 (for practical purposes) to 0.01.

For this piece of the function, x is greater than 0 but not equal to 0.

Press Enter.

Point to its graph in Graphics view.

Its graph is seen in Graphics view.

Point to the blue function.

Point to intersection of f(x) and y-axis at (0,3).

Point to the red function.

Now click on Move Graphics View and drag the background until you can see both graphs.

This is because x is not 0 when f of x is 3 times x plus 1.

When x tends to 0, f of x is 3 as the function intersects the y-axis at 0 comma 3.

When x equals 1, the value of f of x is 6.

Summary

Understand limits of functions

Look at continuity of functions

Assignment

Find the limit of (x³-2x²)/(x²-5x+6) as x tends to 2.

Evaluate lim sin4x

x→0 sin 2x

Find the limit of this rational polynomial function as x tends to 2.

Find the limit of this trigonometric function as x tends to 0.

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