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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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Some examples of functions evaluating to indeterminate values are

`(sin x)/x` when `x=0`

`(e^x-1)/x` when `x=0`

`(f(x+delta)-f(x))/delta` when `delta=0`

It is noted that function provide a mathematical model to applications.

Examples:

Shapes - like circumference of circle

Motion - like speed of a ball thrown

System - like heat emitted in chemical reactions

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 `3`cm: there will be a tiny error; actual radius is `3.00034`cm.

 •  Ball thrown with `2`m/sec velocity : there will be a small error; actual velocity is `1.99948`m/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`. The values close to `x=1` are

at `x=0.999999`

at `x=1.000001`

`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.

The two classes are named as given below.

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`

The terminology used to convey `delta` is close to `0`, but not `0` is: limit delta tending to `0`.

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))`.

familiarize with the terminology
limit

familiarize with the terminology
left hand limit

familiarize with the terminology
right hand limit

                            
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