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Thought-Process to Discover Knowledge

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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

Indeterminate value in functions

Some functions evaluate to indeterminate value at some input values. This is illustrated with an example.



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Consider the value of function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` when `x=2.2`.

On substituting `x=2.2`, we get `f(2.2)= color(deepskyblue)(2.2^2-1)/color(coral)(2.2-1)`.
`f(2.2)=3.2`

Consider the value of function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` when `x=1`

On substituting `x=1`, we get
`f(1)`
`quad quad =color(deepskyblue)(1^2-1)/color(coral)(1-1)`
`quad quad = color(deepskyblue)(0)/color(coral)(0)`

That is, the function evaluates to indeterminate value when `x=1`.

Let us closely examine the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)`.

The numerator be factorized as `x^2-1 = (x+1)(x-1)`

Rewriting the function `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` as the function `f(x) = color(deepskyblue)((x+1)(x-1))/color(coral)(x-1)`.

It is noted that the function can be simplified to `f(x)= x+1` when `x!=1`. Note that `0` cannot be canceled out in expressions or equations.

So the given function
 `f(x)`
 `quad quad = x+1 ` when `x!=1`
 `quad quad = (x^2-1)/(x-1) ` when `x=1`

By this it is concluded that `f(x)|_(x=1)` is indeterminate value `0/0`.

Many students wrongly understand that the algebraic simplification (like canceling `x-1` in the example above) solves the indeterminate value. It is not so -- the function remains indeterminate at that input value `x=1`.

Function evaluates to indeterminate value: Function `f(x)` evaluates to indeterminate value for `x=a` if `f(a) = 0/0`.

Solved Exercise Problem:

Given `f(x)=(x^3-8)/(x^2-4)` What is `f(2)`?

  • `0/0`
  • `0/0`
  • `10`
  • `1`

The answer is '`0/0`'

                            
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