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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

lim x a f ( x ) g ( x ) .

If lim x a f ( x ) = L 1 and lim x a g ( x ) = L 2 0 , then

lim x a f ( x ) g ( x ) = L 1 L 2 .

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

lim x 2 x 2 4 x 2 and lim x 0 sin x x .

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

lim x 2 x 2 4 x 2 = lim x 2 ( x + 2 ) ( x 2 ) x 2 = lim x 2 ( x + 2 ) = 2 + 2 = 4 .

For lim x 0 sin x x we were able to show, using a geometric argument, that

lim x 0 sin 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 lim x a f ( x ) = 0 = lim x a g ( x ) and such that g ( a ) 0 For x near a , we can write

f ( x ) f ( a ) + f ( a ) ( x a )

and

g ( x ) g ( a ) + g ( a ) ( x a ) .

Therefore,

f ( x ) g ( x ) f ( a ) + f ( a ) ( x a ) g ( a ) + g ( a ) ( x a ) .
Two functions y = f(x) and y = g(x) are drawn such that they cross at a point above x = a. The linear approximations of these two functions y = f(a) + f’(a)(x – a) and y = g(a) + g’(a)(x – a) are also drawn.
If lim x a f ( x ) = lim x a g ( x ) , then the ratio f ( x ) / g ( x ) is approximately equal to the ratio of their linear approximations near a .

Since f is differentiable at a , then f is continuous at a , and therefore f ( a ) = lim x a f ( x ) = 0 . Similarly, g ( a ) = lim x a g ( x ) = 0 . If we also assume that f and g are continuous at x = a , then f ( a ) = lim x a f ( x ) and g ( a ) = lim x a g ( x ) . Using these ideas, we conclude that

lim x a f ( x ) g ( x ) = lim x a f ( x ) ( x a ) g ( x ) ( x a ) = lim x a f ( x ) g ( x ) .

Note that the assumption that f and g are continuous at a and g ( a ) 0 can be loosened. We state L’Hôpital’s rule formally for the indeterminate form 0 0 . Also note that the notation 0 0 does not mean we are actually dividing zero by zero. Rather, we are using the notation 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 lim x a f ( x ) = 0 and lim x a g ( x ) = 0 , then

lim x a f ( x ) g ( x ) = lim x a f ( x ) g ( x ) ,

assuming the limit on the right exists or is or . This result also holds if we are considering one-sided limits, or if a = and .

Questions & Answers

what is the power rule
Vanessa Reply
how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
Rodgers Reply
use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C
navin
find the nth differential coefficient of cosx.cos2x.cos3x
Sudhanayaki Reply
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
Crystal Reply
f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x)
Andrew
in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that?
Robert Reply
can you please post the question again here so I can see what your talking about
Andrew
√(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c
Barnabas
564356
Myong
can you determine whether f(x)=x(cube) +4 is a one to one function
Crystal
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor
Robert Reply
step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background!
Andrew
first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2
Ioana
Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x
Melin
thank you ladies and gentlemen I appreciate the help!
Robert
keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be.
Andrew
please how do I learn integration
aliyu Reply
they are simply "anti-derivatives". so you should first learn how to take derivatives of any given function before going into taking integrals of any given function.
Andrew
best way to learn is always to look into a few basic examples of different kinds of functions, and then if you have any further questions, be sure to state specifically which step in the solution you are not understanding.
Andrew
example 1) say f'(x) = x, f(x) = ? well there is a rule called the 'power rule' which states that if f'(x) = x^n, then f(x) = x^(n+1)/(n+1) so in this case, f(x) = x^2/2
Andrew
great noticeable direction
Isaac
limit x tend to infinite xcos(π/2x)*sin(π/4x)
Abhijeet Reply
can you give me a problem for function. a trigonometric one
geovanni Reply
find volume of solid about y axis and y=x^3, x=0,y=1
amisha Reply
Practice Key Terms 2

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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