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Thought-Process to Discover Knowledge

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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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What is the value of `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` when `x=2.2`?

  • `0/0`
  • indeterminate value
  • `3.2`
  • `3.2`

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

What is the value of `f(x) = color(deepskyblue)(x^2-1)/color(coral)(x-1)` when `x=1`?

  • `0/0`
  • indeterminate value
  • both the above
  • both the above

The answer is 'Both the above'. On substituting `x=1`, we get
`quad quad =color(deepskyblue)(1^2-1)/color(coral)(1-1)`
`quad quad = color(deepskyblue)(0)/color(coral)(0)`.

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

  • Yes, `x^2-1 = (x+1)(x-1)`
  • Yes, `x^2-1 = (x+1)(x-1)`
  • Cannot be factorized as Factorization is only for numbers, not for algebraic expressions

The answer is 'Yes - `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)`. Can the function be simplified?

  • Simplified to `f(x)= x+1`
  • Simplified only when `x!=1` to `f(x)= x+1`
  • Simplified only when `x!=1` to `f(x)= x+1`

The answer is 'Simplified to `f(x)= x+1` when `x!=1`'. Note that `0` cannot be canceled out in expressions or equations.

So the given function
 `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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