# 4.8 L’hôpital’s rule

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• Recognize when to apply L’Hôpital’s rule.
• Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
• Describe the relative growth rates of functions.

In this section, we examine a powerful tool for evaluating limits. This tool, known as L’Hôpital’s rule    , uses derivatives to calculate limits. With this rule, we will be able to evaluate many limits we have not yet been able to determine. Instead of relying on numerical evidence to conjecture that a limit exists, we will be able to show definitively that a limit exists and to determine its exact value.

## Applying l’hôpital’s rule

L’Hôpital’s rule can be used to evaluate limits involving the quotient of two functions. Consider

$\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}.$

If $\underset{x\to a}{\text{lim}}f\left(x\right)={L}_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\underset{x\to a}{\text{lim}}g\left(x\right)={L}_{2}\ne 0,$ then

$\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{{L}_{1}}{{L}_{2}}.$

However, what happens if $\underset{x\to a}{\text{lim}}f\left(x\right)=0$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=0?$ We call this one of the indeterminate forms    , of type $\frac{0}{0}.$ This is considered an indeterminate form because we cannot determine the exact behavior of $\frac{f\left(x\right)}{g\left(x\right)}$ as $x\to a$ without further analysis. We have seen examples of this earlier in the text. For example, consider

$\underset{x\to 2}{\text{lim}}\frac{{x}^{2}-4}{x-2}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{x}.$

For the first of these examples, we can evaluate the limit by factoring the numerator and writing

$\underset{x\to 2}{\text{lim}}\frac{{x}^{2}-4}{x-2}=\underset{x\to 2}{\text{lim}}\frac{\left(x+2\right)\left(x-2\right)}{x-2}=\underset{x\to 2}{\text{lim}}\left(x+2\right)=2+2=4.$

For $\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{x}$ we were able to show, using a geometric argument, that

$\underset{x\to 0}{\text{lim}}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{x}=1.$

Here we use a different technique for evaluating limits such as these. Not only does this technique provide an easier way to evaluate these limits, but also, and more important, it provides us with a way to evaluate many other limits that we could not calculate previously.

The idea behind L’Hôpital’s rule can be explained using local linear approximations. Consider two differentiable functions $f$ and $g$ such that $\underset{x\to a}{\text{lim}}f\left(x\right)=0=\underset{x\to a}{\text{lim}}g\left(x\right)$ and such that ${g}^{\prime }\left(a\right)\ne 0$ For $x$ near $a,$ we can write

$f\left(x\right)\approx f\left(a\right)+{f}^{\prime }\left(a\right)\left(x-a\right)$

and

$g\left(x\right)\approx g\left(a\right)+{g}^{\prime }\left(a\right)\left(x-a\right).$

Therefore,

$\frac{f\left(x\right)}{g\left(x\right)}\approx \frac{f\left(a\right)+{f}^{\prime }\left(a\right)\left(x-a\right)}{g\left(a\right)+{g}^{\prime }\left(a\right)\left(x-a\right)}.$

Since $f$ is differentiable at $a,$ then $f$ is continuous at $a,$ and therefore $f\left(a\right)=\underset{x\to a}{\text{lim}}f\left(x\right)=0.$ Similarly, $g\left(a\right)=\underset{x\to a}{\text{lim}}g\left(x\right)=0.$ If we also assume that ${f}^{\prime }$ and ${g}^{\prime }$ are continuous at $x=a,$ then ${f}^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}{f}^{\prime }\left(x\right)$ and ${g}^{\prime }\left(a\right)=\underset{x\to a}{\text{lim}}{g}^{\prime }\left(x\right).$ Using these ideas, we conclude that

$\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\text{lim}}\frac{{f}^{\prime }\left(x\right)\left(x-a\right)}{{g}^{\prime }\left(x\right)\left(x-a\right)}=\underset{x\to a}{\text{lim}}\frac{{f}^{\prime }\left(x\right)}{{g}^{\prime }\left(x\right)}.$

Note that the assumption that ${f}^{\prime }$ and ${g}^{\prime }$ are continuous at $a$ and ${g}^{\prime }\left(a\right)\ne 0$ can be loosened. We state L’Hôpital’s rule formally for the indeterminate form $\frac{0}{0}.$ Also note that the notation $\frac{0}{0}$ does not mean we are actually dividing zero by zero. Rather, we are using the notation $\frac{0}{0}$ to represent a quotient of limits, each of which is zero.

## L’hôpital’s rule (0/0 case)

Suppose $f$ and $g$ are differentiable functions over an open interval containing $a,$ except possibly at $a.$ If $\underset{x\to a}{\text{lim}}f\left(x\right)=0$ and $\underset{x\to a}{\text{lim}}g\left(x\right)=0,$ then

$\underset{x\to a}{\text{lim}}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\text{lim}}\frac{{f}^{\prime }\left(x\right)}{{g}^{\prime }\left(x\right)},$

assuming the limit on the right exists or is $\infty$ or $\text{−}\infty .$ This result also holds if we are considering one-sided limits, or if $a=\infty \phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}-\infty .$

#### Questions & Answers

why n does not equal -1
K.kupar Reply
ask a complete question if you want a complete answer.
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
Darnell Reply
proof the formula integration of udv=uv-integration of vdu.?
Bg Reply
Find derivative (2x^3+6xy-4y^2)^2
Rasheed Reply
no x=2 is not a function, as there is nothing that's changing.
Vivek Reply
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
Joys Reply
y=800
Gift
800
Bg
how do u factor the numerator?
Drew Reply
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Kranthi Reply
answer please?
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Lerato Reply
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
Amna Reply
what is a function? f(x)
Jeremy Reply
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
jon Reply
is x=2 a function?
The
What is limit
MaHeSh Reply
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
Andrew Reply
what is implicit of y³+x²y³=5 at (2,1)
Estelita Reply
tel mi about a function. what is it?
Jeremy
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place. Example A son depends on his father. meaning here is the father is function of the son. let the father be y and the son be x. the we say F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable
moe
x or y those not matter is just to represent.
Bg

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