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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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mathsLimit of a functionUnderstanding limits with Graphs

### L'Hospital's Rule

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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