# L'Hopital Rule

• Jul 18th 2012, 11:02 PM
TWN
L'Hopital Rule
1.) What is the limit of ( e^x - (1+x) ) / x^n as x approaches 0 from the right?

Using the L'Hopital rule, I'm a little confused with regard to the denominator.

2.) What is the limit of (squareroot(25-x^2)) / (x^2 - 1) when x approaches 5 from the left?

• Jul 18th 2012, 11:39 PM
pratique21
Re: L'Hopital Rule
Hi!
Fir the first one you need to apple L'Hospital recursively until you rid the 'x' term in the denominator

(ex - (1+x)) / xn
so differentiate to get
(ex - 1 ) / n * xn-1
Repeat
(ex) / n(n-1) * xn-2

So now you know that if you keep differentiating, the numerator remains unaffected. However the denominator will reduce to something like
(n)(n-1)(n-2)......1 (= say N)

Now put the limiting value of x=0
you get
1/ N

Regarding the second one, i think the left hand limit will come out to be 0. Since the domain of the function is (-inf,5] , the LHL exists as x->5-

Hope it helps !
• Jul 18th 2012, 11:39 PM
Prove It
Re: L'Hopital Rule
Quote:

Originally Posted by TWN
1.) What is the limit of ( e^x - (1+x) ) / x^n as x approaches 0 from the right?

Using the L'Hopital rule, I'm a little confused with regard to the denominator.

2.) What is the limit of (squareroot(25-x^2)) / (x^2 - 1) when x approaches 5 from the left?

Surely you know that the derivative of x^n is nx^(n-1)...
• Jul 18th 2012, 11:40 PM
Prove It
Re: L'Hopital Rule
Quote:

Originally Posted by pratique21
Hi!
Fir the first one you need to apple L'Hospital recursively until you rid the 'x' term in the denominator

(ex - (1+x)) / xn
so differentiate to get
(ex - 1 ) / n * xn-1
Repeat
(ex) / n(n-1) * xn-2

So now you know that if you keep differentiating, the numerator remains unaffected. However the denominator will reduce to something like
(n)(n-1)(n-2)......1 (= say N)

Now put the limiting value of x=0
you get
1/ N

Regarding the second one, i think the left hand limit will come out to be 0. Since the domain of the function is (-inf,5] , the LHL exists as x->5-

Hope it helps !

Isn't n(n-1)(n-2)...(3)(2)(1) = n! ?
• Jul 18th 2012, 11:49 PM
pratique21
Re: L'Hopital Rule
Yeah. 1/n!
Missed that. Thanks ! ;)
• Jul 18th 2012, 11:57 PM
pratique21
Re: L'Hopital Rule
Quote:

Originally Posted by Prove It
Isn't n(n-1)(n-2)...(3)(2)(1) = n! ?

Just btw, if we expand ex and then see, we have
1+x+x2/2! + x3/3! + ...... xn/n! + ..... to inf

so our expression becomes

(1+x+x2/2! + x3/3! + ...... xn/n! + ..... to inf - (1+x) )/ xn
= (x2/2! + x3/3! + ...... xn/n! + ..... to inf ) / xn

now for terms beyond xn we need not worry as the higher powers will go to 0 as x-> 0
however if we are to find the limit by dividing the rest of the terms :-
(x2/2! + x3/3! + ...... xn/n!) / xn
for x-> 0
a number of undefined terms will be encountered, apart from 1/n! which seems to be coming if we can somehow deal with the lower powers !

Kindly clarify this !
thanks.
:)