L'Hospital's Rule

L'Hospital's Rule

If f(x)=fn(x)fd(x)f(x)=fn(x)fd(x), where

f(x)∣x=a=00f(x)∣x=a=00;

fn(x)∣x=a=0fn(x)∣x=a=0 and

fd(x)∣x=a=0fd(x)∣x=a=0, then

limx→af(x)limx→af(x)
=[ddxfn(x)]∣x→a =[ddxfn(x)]∣∣∣x→a
÷[ddxfd(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.*

introduction

The function f(x)=fn(x)fd(x) such that fn(x)∣x=a=0 and fd(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.

formal handling with slop

Given the function f(x)=fn(x)fd(x) such that fn(x)∣x=a=0 and fd(x)∣x=a=0.

f(x)∣x=a+δ

=fn(x)|_(x=a+δ)÷fd(x)|x=a+δ

=[fn(x)∣x=a+δ-fn(a)]

÷[fd(x)∣x=a+δ-fd(a)]

as fn(a)=0 and fd(a)=0.

=fn(x)∣x=a+δ-fn(a)δ

÷fd(x)∣x=a+δ-fd(a)δ

=slopefn(x)∣x=a÷slopefd(x)∣x=a.

The beauty of this mathematical manupulation is that the function evaluated at a position is 00 which is not deterimined to be a value. The same is equivalently given as slope of numerator divided by slope of dinominator which provides another way to determine the value.

revisit

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.

example

Given function f(x)=x2-1x-1. what is limx→1f(x)?

The answer is '2'.

Differentiating numerator ddx(x2-1)∣x=1=2

Differentiating denominator ddx(x-1)∣x=1=1

limx→1f(x)

=21

summary

**L'Hospital's Rule: **If f(x)=fn(x)fd(x), where

f(x)∣x=a=00;

fn(x)∣x=a=0 and

fd(x)∣x=a=0, then

limx→af(x)

=[ddxfn(x)]∣x→a

÷[ddxfd(x)]∣x→a

when the numerator and denominator are differentiable.

Outline

The outline of material to learn "limits (calculus)" is as follows.

Note : * click here for detailed outline of Limits(Calculus).*

→ __Indeterminate and Undefined__

→ __Indeterminate value in Functions__

→ __Expected Value__

→ __Continuity__

→ __Definition by Limits__

→ __Geometrical Explanation for Limits__

→ __Limit with Numerator and Denominator__

→ __Limits of Ratios - Examples__

→ __L'hospital Rule__

→ __Examining a function__

→ __Algebra of Limits__

→ __Limit of a Polynomial__

→ __Limit of Ratio of Zeros__

→ __Limit of ratio of infinities__

→ __limit of Binomial__

→ __Limit of Non-algebraic Functions__