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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.

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The function, for which limit is computed, is considered as two constituent functions of numerator and denominator. To find the limit of the function, differentiate the numerator and denominator.



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The function `f(x)=color(deepskyblue)(f_n(x))/color(coral)(f_d(x))` such that `color(deepskyblue)(f_n(x))|_(x=a) = 0` and `color(coral)(f_d(x))|_(x=a) = 0`. It was discussed that the slope of the numerator and denominator defines the limits. This is formally given by L'Hospital's Rule.

There are multiple proofs for L'Hospital's Rule. The discussion on slopes here is the intuitive understanding (not a formal proof) of L'Hospital's Rule.

Given the function `f(x)=color(deepskyblue)(f_n(x))/color(coral)(f_d(x))` such that `color(deepskyblue)(f_n(x))|_(x=a) = 0` and `color(coral)(f_d(x))|_(x=a) = 0`.

`f(x)|_(x=a+delta)`
`quad quad = color(deepskyblue)(f_n(x)) |_(x=a+delta) -: color(coral)(f_d(x))|_(x=a+delta)`
`quad quad = [color(deepskyblue)(f_n(x))|_(x=a+delta) - f_n(a)]`
`quad quad quad quad -: [color(coral)(f_d(x))|_(x=a+delta) - f_d(a)]`
as `f_n(a) = 0` and `f_d(a)=0`.

`quad quad = [color(deepskyblue)(f_n(x))|_(x=a+delta) - f_n(a)]/delta`
`quad quad quad quad -: [color(coral)(f_d(x))|_(x=a+delta) - f_d(a)]/delta`
`quad quad = color(deepskyblue)(text(slope) f_n(x)|_(x=a)) -: color(coral)(text(slope) f_d(x)|_(x=a))`.

If you have started on the calculus and limits, then you may not have come across derivative, differentiation, and differentiability. If required, you may have to revisit this page when you have completed the differential calculus.

For the limit of a function, evaluate the function formed by derivatives of the numerator and the denominator.

L'Hospital's Rule: If `f(x)=color(deepskyblue)(f_n(x))/color(coral)(f_d(x))`, where
`f(x)|_(x=a) = 0/0`;
`color(deepskyblue)(f_n(x))|_(x=a) = 0` and
`color(coral)(f_d(x))|_(x=a) = 0`, then
`lim_(x->a) f(x) `
`quad quad = color(deepskyblue)([d/(dx) f_n(x)]|_(x->a)) `
`quad quad quad quad -: color(coral)([d/(dx) f_d(x)]|_(x->a))`
when the numerator and denominator are differentiable.

Solved Exercise Problem:

Given function `f(x)=color(deepskyblue)(x^2-1)/color(coral)(x-1)`. what is `lim_(x->1)f(x)`?

  • `2`
  • `2`
  • `1`
  • `0/0`

The answer is '`2`'.

Differentiating numerator `d/(dx) color(deepskyblue)((x^2-1)|_(x=1) = 2)`
Differentiating denominator `d/(dx) color(coral)((x-1)|_(x=1) = 1)`

`lim_(x->1)f(x)`
`quad quad = 2/1`

                            
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