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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsLimit of a functionBasics: Limit of a function

Continuity of a Function at an input value

Even when functions have a defined value at an input value, the function may be discontinuous. Limit of the function at that input value can be used to determine continuity.



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So far, the motivation to examine limits of a function was to evaluate the function at a input value of the argument variable where the function evaluates to 'indeterminate value'.
In this topic, another motivation to examine limits is explained.

Consider `f(x)=1/color(coral)(x-1)`. What is `f(1)`?

  • `+oo`
  • `+oo`
  • `-oo`

The answer is '`+oo`', by directly substituting `x=1`.

Consider `f(x)=1/color(coral)(x-1)`. What is `lim_(x->1+)f(x)`?

  • `+oo`
  • `+oo`
  • `-oo`

The answer is '`oo`', because
 `lim_(x->1+)f(x)`
 `quad quad = 1/color(coral)(1+delta-1) `
 `quad quad = 1/color(coral)(delta) = 1/0 = oo`

Consider `f(x)=1/color(coral)(x-1)`. What is `lim_(x->1-)f(x)`?

  • `+oo`
  • `-oo`
  • `-oo`

The answer is '`-oo`', because
 `lim_(x->1-)f(x)`
 `quad quad = 1/color(coral)(1-delta-1) `
 `quad quad = 1/color(coral)(-delta)`
 `quad quad = -1/0 = -oo`

For the function `f(x)=1/color(coral)(x-1)`,
 •  `f(a) = oo`
 •  `lim_(x->1+)f(x) = oo`
 •  `lim_(x->1-)f(x) = -oo`

The plot of the function is given in the figure.graph of function 1/(x-1) What does this tell about the function at `x=1`?

  • function is not continuous
  • function is not continuous
  • does not have any significance

The answer is 'function is not continuous'.
For a value less than `x=1`, the function is `-oo`. And at `x=1`, the function becomes `oo`

A function `f(x)` at a given input value `x=a` is continuous if all the three are equal
`f(x)|_(x=a)`
`quad quad = lim_(x->a-) f(x)`
`quad quad = lim_(x->a+) f(x)`

What does the word 'continuous' mean?

  • unbroken and continue from one side to another without pause in between
  • unbroken and continue from one side to another without pause in between
  • broken into pieces

The answer is 'unbroken and continue from one side to another without pause in between'.

What is the term used to describe a function at an input value, if value of the function, left-hand-limit, and right-hand-limit are equal ?

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is 'continuous'

A function is continuous at an input value, if the function evaluated at the input equals both left-hand limit and right-hand limit of the function at that input value.

Continuity of a Function: A function `f(x)` is continuous at `x=a` if all the following three have a defined value and are equal

 •  Evaluated at the input value `f(x)|_(x=a)`

 •  left-hand-limit `lim_(x->a-) f(x)`

 •  right-hand-limit `lim_(x-a+) f(x)`

Solved Exercise Problem:

Given function `f(x) = 2x^2`, is it continuous at x=0?

  • Yes, Continuous.
  • Yes, Continuous.
  • No, Not continous

The answer is 'Yes, Continuous'. Evaluate the three values of the function and they are equal.

                            
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