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Basics: Limit of a function

Basics: Limit of a function

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Expected Value of Functions


 »  A function evaluate to `0/0` at `x=a`
    →  Considering that functions model real world quantities, in setting `x=a` there can be very small error.

    →  Considering functions in abstraction, the value at `x=a` can be assumed to be that of `x=a+- delta` with `delta` close to `0`


 »  If a function is evaluated to `0/0` at `x=a`
    →  expected value can be that of `x=a-delta` : left-hand-limit
    →  expected value can be that of `x=a+delta` : right-hand-limit


 »  `f(x)=((x+2)(x-2))/(x-2)`
    →  `x-2` can be canceled for all values of `x` except for `x=2` (as `0` cannot be canceled).
    →  function at value `f(x)|_(x=2) = 0/0` :
    →  left hand limit `lim_(x->2-) f(x) = 2-delta+2 = 4`
    →  right hand limit `lim_(x->2+) f(x) = 2+delta+2 = 4`

Expected Value of a Function

plain and simple summary

nub

plain and simple summary

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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

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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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Starting to learn about 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.

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

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

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 `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`. What are the values close to `x=1`?

  • `x=0.999999`
  • `x=1.000001`
  • 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

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`
  • 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'.

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

  • Practice Saying the Answer

The answer is 'limit'

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

  • Practice Saying the Answer

The answer is 'left hand limit'

What is the term used to convey delta is close to `a` on the right sode of the value, but not the value `a`?

  • Practice Saying the Answer

The answer is 'right hand limit'

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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))`.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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Progress

Can you identify the functions evaluating to indeterminate value in the following?
sine
sine x by x when x equals 0
e;power
e power x minus 1 by x when x = 0
delta
f of x plus delta minus f of x, divided by delta when delta = 0
all;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
Shapes - like circumference of circle
motion
Motion - like speed of a ball thrown
system
system - like heat emitted in chemical reactions
all;above
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 by 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 of 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 centimeter : there will be a tiny error; actual radius is 3 point 0 0 0 3 4 centimeter ;; Ball thrown with 2 meter per second velocity : there will be a small error; actual velocity is 1 point 9 9 9 4 8 meter per second ;; 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 of 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 of x has indeterminate value at x=a. To evaluate the possible value of f of x at x equals a, we examine values close to x=a.
f of x equals x squared minus 1 by x minus 1 evaluates to indeterminate value at x = 1. What are the values close to x = 1?
0 point;9
x = 0 point 9 9 9 9 9 9
1 point; 1.0;
x = 1 point 0 0 0 0 0 1
both;above
both the above
The answer is 'both the above'
f of x equals x squared minus 1 by x minus 1 evaluates to indeterminate value at x=1. But ;; f of 1 point 1 = 2 point 1 ;; f of 0 point 9 = 1 point 9 ;; f of 1 point 0 0 1 = 2 point 0 0 1 ;; f 0f 0 point 9 9 9 9 9 = 1 ppint 9 9 9 9 9 ;; Note that as the value of x is closer to 1, the f of x is closer to 2.
f of x equals x squared minus 1 by x minus 1 evaluates to indeterminate value at x=1. The values closer to x=1 can be either smaller or larger. 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
left-side of the value
right
right-side of the value
both;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 of x evaluates to indeterminate value at x=a. To evaluate the expected value of f of x at x = a, we examine values close to x=a ;; at right-hand-side x=a+delta ;; at left-hand-side x=a minus delta
f of x equals x squared minus 1 by x minus 1 evaluates to indeterminate value at x=1. Evaluating right hand side at x = 1+delta ;; f of 1 +delta ;; equals 1+ delta squared minus 1, by, 1 + delta minus 1 ;; equals 1 + 2 delta + delta squared minus 1, by, delta;; equals 2 plus delta, multiplied delta divided by delta;; equals 2 + delta ;; equals 2 ;; as delta is approximately 0 ;; Note that in one of the steps delta is canceled. That is not same as canceling 0 as delta is approximately 0. At the same time, in the last step delta is substituted as 0, as delta is negligibly small compared to 2.
f of x equals x squared minus 1 by x minus 1 evaluates to indeterminate value at x=1. Evaluating left hand side at x = 1 minus delta ;; f of 1 minus delta ;; equals 1 minus delta squared minus 1, by, 1 minus delta minus 1 ;; equals 1 minus 2 delta + delta squared minus 1, by, minus delta;; equals 2 minus delta, multiplied minus delta divided by minus delta;; equals 2 minus delta ;; equals 2 ;; as delta is approximately 0
Can you guess the terminology used to convey delta is close to 0 , but not 0 ?
limit
limit delta to 0
grand;value
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 of x evaluates to indeterminate value at x=a. To evaluate the possible value of f of x at x = a, The function is examined ;; at right-hand-side x=a+delta given as limit x tending to a. plus f of x ;; at left-hand-side x=a minus delta given as limit x tending to a. minus f of x
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 of x, the expected values of the function at x=a are ;; Right-hand-limit limit : x tending to a. plus f of x; equals; limit delta tending to 0 f of a plus delta;; and left-hand-limit : limit x tending to a. minus f of x; equals; limit delta tending to 0 f of a minus delta
What is the term used to convey delta is close to a , but not equal to a ?
limit
The answer is 'limit'
What is the term used to convey delta is close to a on the left side of the value, but not the value a ?
left hand limit
The answer is 'left hand limit'
What is the term used to convey delta is close to a on the right sode of the value, but not the value a ?
right hand limit
The answer is 'right hand limit'

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