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Understanding limits with Graphs

Understanding limits with Graphs

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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`.

L'Hospital's Rule

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

trek

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.


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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

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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.

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