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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

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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 -: color(coral)([d/(dx) f_d(x)]|_(x->a))`

when the numerator and denominator are differentiable.*Note: The limit is the slope of numerator divided by slope of denominator at `x=a`.*

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

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

*simple steps to build the foundation*

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*simple steps to build the foundation*

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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.

Starting on learning "lopitals 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.

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.

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

- `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`

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**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.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

*your progress details*

Progress

*About you*

Progress

The function f of x = f n of x divided by f d of x;; where f n of x at x=a, = 0 ;; and f d of x at x=1, = 0 . ;; it was discussed that the slope of the numerator and denominator defines the limits. This is formally given by lopitals rule;; There are multiple mathematically rigorous proofs for lopitals rule. The discussion on slopes here is the intuitive understanding, not a formal proof, of lopitals rule.

Given the function f of x = f n of x by f d of x such that, f n of x at x = a, = 0, and f_d of x at x = a, = 0. ;; f of x at x=a+delta ;; equals f n of x at x=a+delta divided by f d of x at x=a+delta ;; equals f n of x at x=a+delta, minus f n of a, divided by, f d of x at x=a+delta, minus f d of a ;; as f n of a and f d of a are 0 ;; equals f n of x at x=a+delta, minus f n of a, by delta, divided by, f d of x at x=a+delta, minus f d of a, by delta ;; equals slope of f n of x at x=a, divided by, slope of f d of x at x=a;

.

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

lopitals rule : If f of x = f n of x divided by f d of x, where ;; f of x at x=a, = 0 by 0 ;; f n of x at x = a, = 0 ;; f d of x at x = a, = 0;; then limit x tending to a f of x ;; equals derivative of f n of x evaluated at x = a, divided by derivative of f d of x evaluated at x = a. This is valid when the numerator and the denominator are differentiable.

Given function f of x = x squared minus 1 by x minus 1, what is limit x tending to 1 f of x ?

2

2

1

1

0

0 by 0

The answer is "2". Differentiating the numerator : d by d x of x squared minus 1, at x=1, = 2 ;; differentiating the denominator d by d x of x minus 1, at x=1, = 1 ;; limit x tending to 1 f of x equals 1 by 1.