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Thought-Process to Discover Knowledge

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mathsLimit of a functionBasics: Limit of a function

Expected Value of a Function

When a function evaluate to indeterminate value, the limits are computed at that input value. Two limits, left-hand-limit and right-hand-limit are introduced and explained.

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Can you identify the functions evaluating to indeterminate value in the following?

• (sin x)/x when x=0
• (e^x-1)/x when x=0
• (f(x+delta)-f(x))/delta when delta=0
• all the above
• all the above

The answer is 'All the above'

What are the practical applications where functions can provide a mathematical model to the application?

• Shapes - like circumference of circle
• Motion - like speed of a ball thrown
• system - like heat emitted in chemical reactions
• All the above and many more
• All the above and many more

The answer is 'All the above and many more'.

In such applications we come across functions that have indeterminate values at certain values for the argument variable. The motivation in applied mathematics is to solve such functions having indeterminate values. In such systems, setting a parameter to an absolute value is not practically possible. This is examined further in following pages.

The motivation in abstract mathematics is to find what is the expected value of the function where it has indeterminate value. It is known that indeterminate value 0/0 possibly takes a value depending on the problem. Can that be determined?

Note: Apart from evaluating a function at an input value, there is another motivation to the theory being discussed. It is covered in the topic on 'continuity of functions'.

Given that function f(x) evaluates to indeterminate value at x=a. The approach to finding the expected value is
•  If this represents an application with some parameter x=a, the quantity x cannot be exactly set to value a in practical situations. There will be a small error to value of x.

•  circle of radius 3cm: there will be a tiny error; actual radius is 3.00034cm.

•  Ball thrown with 2m/sec velocity : there will be a small error; actual velocity is 1.99948m/sec

•  If it is abstract, is the function defined at input values very close to x=a? Then those values can be used to find expected value of f(a)

If a function evaluates to indeterminate value at an input value, then examine the nearby input values close to the input value.

Given that function f(x) has indeterminate value at x=a. To evaluate the possible value of f(x)|_(x=a), we examine values close to x=a.

f(x) = (x^2-1)/(x-1) evaluates to indeterminate value at x=1. What are the values close to x=1?

• x=0.999999
• x=1.000001
• both the above
• both the above

The answer is 'both the above'

f(x) = (x^2-1)/(x-1) evaluates to indeterminate value at x=1. But
f(1.1) = 2.1
f(0.9) = 1.9
f(1.001) = 2.001
f(0.99999) = 1.99999
Note that as the value of x is closer to 1, the f(x) is closer to 2.

f(x) = (x^2-1)/(x-1) evaluates to indeterminate value at x=1.

The values closer to x=1 can be either

•  smaller : .9999, .9999999, etc.

•  larger : 1.0001, 1.0000001, etc.

This makes two classes of input values for which the function is evaluated.

What could be a good name to refer to these two classes?

• left-side of the value
• right-side of the value
• both the above
• both the above

The answer is 'both the above'.

The values smaller than x=1 is referred as 'left-hand-side' and The values larger than x=1 is referred as 'right-hand-side'.

Given that function f(x) evaluates to indeterminate value at x=a. To evaluate the expected value of f(x)|_(x=a), we examine values close to x=a;

•  at right-hand-side x=a+delta

•  at left-hand-side x=a-delta

f(x) = (color(deepskyblue)(x^2-1))/(color(coral)(x-1)) evaluates to indeterminate value at x=1. Evaluating right-hand-side at x=1+delta

f(1+delta)
quad quad = (color(deepskyblue)((1+delta)^2-1))/(color(coral)(1+delta-1))
quad quad = (color(deepskyblue)(1+2delta+delta^2-1))/(color(coral)(delta))
quad quad = (color(deepskyblue)((2+delta)delta))/(color(coral)(delta))
quad quad = color(deepskyblue)(2+delta)
quad quad = 2 quad quad quad quad  as delta ~=0

Note that in one of the steps delta is canceled. That is not same as canceling 0 as the delta ~=0. At the same time, in the last step delta is substituted as 0, as delta is negligibly small compared to 2.

f(x) = (x^2-1)/(x-1) evaluates to indeterminate value at x=1. Evaluating left-hand-side at x=1-delta

f(1-delta)
quad quad = (color(deepskyblue)((1-delta)^2-1))/(color(coral)(1-delta-1))
quad quad = (color(deepskyblue)(1-2delta+delta^2-1))/(color(coral)(-delta))
quad quad = (color(deepskyblue)((2-delta)(-delta)))/(color(coral)(-delta))
quad quad = color(deepskyblue)(2-delta)
quad quad = 2 quad quad quad quad as delta ~=0

Can you guess the terminology used to convey delta is close to 0, but not 0?

• limit delta to 0
• limit delta to 0
• grand value of delta is 0

The answer is 'limit delta to 0'. Mathematicians use the word 'limit' to convey the meaning.

Given that function f(x) evaluates to indeterminate value at x=a. To evaluate the possible value of f(x)|_(x=a), The function is examined

•  at right-hand-side x=a+delta given as lim_(x->a+)f(x)

•  at left-hand-side x=a-delta given as lim_(x->a-)f(x)

Note: lim_(x->a+) is read as 'limit x tending to a plus'.

When a function evaluates to indeterminate value, expected values or limits at that input value is calculated for values very close to that input value.
Two limits are defined left-hand-limit and right-hand-limit.

Right-hand-Limit and Left-hand-limit: Given function f(x) the expected values of the function at x=a are
Right-hand-limit color(deepskyblue)(lim_(x->a+) f(x)=lim_(delta->0) f(a+delta)) and
Left-hand-limit color(coral)(lim_(x->a-) f(x)=lim_(delta->0) f(a-delta)).

What is the term used to convey delta is close to a, but not equal to a?

• Pronunciation : Say the answer once

What is the term used to convey delta is close to a on the left side of the value, but not the value a?

• Pronunciation : Say the answer once
What is the term used to convey delta is close to a on the right sode of the value, but not the value a?