What is the L'Hopital's rule and how do you use it?

Question

What is the L'Hopital's rule and how do you use it?

Examples would be very helpful

Answer 1

First Rule Example:
Find the value of (Lim x tends to 0) Sin x / x
If we plug x = 0,
Sin x / x = Sin 0 / 0 = 0 / 0 = indeterminate / Not defined number

L'Hopital's rule: differentiate numerator and denominator individully and plug the value again. If the value is again 0 / 0, repeat the process and so on.

[d/dx(Sin x)] / [d/dx(x)] = Cos x / 1 = Cos x
Plug x = 0,
Cos x = Cos 0 = 1
So (Lim x tends to 0) Sin x / x = 1

Second Rule Example:
Find the value of (lim x tends to infinity) (x^2 + x + 222) / (2x^2 + 12x + 1000)
If we plug x = infinity,
(x^2 + x + 222) / (2x^2 + 12x + 1000) = infinity / infinity = undefined number

L'Hopital's rule: differentiate numerator and denominator individully and plug the value again. If the value is again infinity/infinity, repeat the process and so on.

Differentiating numerator and denominator
(2x +1) / (2.2x + 12) = (2x + 1) / (4x + 12)
Plugging x = infinity will result as infinity / infinity again.
Differentiating again,
(2x1 + 0) / (4x1 + 0) = 2/4 = 1/2

(lim x tends to infinity) (x^2 + x + 222) / (2x^2 + 12x + 1000) = 1/2

Make sure to test that the limiting value of x (independent variable) will produce a value of 0/0 or infinity/infinity for the given function (dependent variable) to be evaluated,
only then,
you can apply the L'Hospital's Rule.

Answer 2

L"Hopital's Rule (pronounced LO-pee-tall) is cool! It allows you to define division by zero in certain circumstances. It is named after Guillaume F. A. DeL'Hopital, who published the first calculus textbook in 1696. It is a particular case of the Extended Mean Value Theorem, which states that, under the conditions that follow,
f'(c)/g'(c) =( f(b)-f(a))/(g(b)-g(a))
(f and g differentiable on (a,b) and continuous on [a,b]. g(x) not equal to zero for any x in (a,b), we can find a c in (a,b) such that this statement is true.)
L'Hopital's rule states that if lim((f(x)/g(x)) is in the indeterminante form 0/0 or plus or minus infinity / plus or minus infinity, then this limit is equal to lim (f'(x)/g'(x)).
In other words, you simply take the derivative of each function (separately, not by the quotient rule) and compare these derivatives. You can do this as many times as necessary, as long as you keep coming up with the appropriate indeterminate forms 0/0 or infinity / infinity.
Example
lim x ->0 ((e^(2x) -l)/x)
By inspection and direct substitution, this is 0/0. So take the derivative of top and bottom and you get
lim x->0 (2e^(2x)/1) which is 2

(source: calculus textbook, -Calculus with Analytic Geometry- by Larson and Hostetler)

Answer 3

See previous post

If you have a ratio f(x)/g(x) and that ratio is an "indeterminate form" (like 0/0 or infinity/infinity) then the limit equals the limit of f'(x)/g'(x). Differentiation is like canceling powers of x in this setting.

For instance, limit x -> 0 of sin(x)/x. If you plug in x=0 you get 0/0. So you differentiate top and bottom (separately! NOT the quotient rule!!!!) to get

sin(x) -> cos(x)
x -> 1

So limit sin(x)/x = limit cos(x)/1 = limit cos(x) = cos(0) = 1

For this example it is important that x -> 0: if x -> pi/2 then the limit would be (of course) sin(pi/2)/[pi/2] = 2/pi

Answer 4

L' Hospital Rule: If the fractional function
f(x)/g(x) assumes one of the indeterminate forms 0/0 or infinity/infinity (where a is finite or infinite), then
limit f(x)/g(x) equal to the first of the expressions
x tends to a

limit f'(x)/g'(x); Limit f"(x)/ g"(x) ; Lt f'"(x)/g'"(x)
tends to a x tends to a x tends to a

which is not indeterminate, provided such first indicated limit exists.

Answer 5

Isn't this the kind of question which is easy to look up in a calculus book, or, dare I say it, on the internet?

You need to learn how to look things up yourself. You won't always be able to rely on obtaining answers from internet forums. This may sound like harsh criticism, but it is truly in your best interest.

Answer 6

It's a method for finding answers to limit problems when they give indeterminite results.
for f(x) & g(x) if f(0)=g(0)=0

if lim x--->0 f(x)/g(x) is indeterminate.
L-Hospital's rule says that it is
lim x--->0 f'(x)/g'(x)
example
f(x)=e^x-1
g(x)=x
lim x--->0 f(x)/g(x)=(1-1)/0=indeterminite
f'(x)=e^x
g'(x)=1
lim x-->0 f'(x)/g'(x)=e^0 /1=1

Answer 7

lim { x cosx / sin2x} x>0 = lim { x cosx / 2sinx*cosx} x>0 =lim { x / 2sinx} x>0 now differentiate numerator and denominator one after the different, u have =lim { a million / 2cosx} x>0 now positioned x=0 answer is a million/2

Answer 8

Study the website under "Source(s)".