Find the limit of the sequence as n tends to 1

In summary, the task is to find the limit of a given sequence as n tends to 1. After applying L'hopital's rule twice, the correct answer is 1/2.
  • #1
sara_87
763
0

Homework Statement



Find the limit of the sequence as n tends to 1

(3/(1-sqrt(x)) - (2/(1-cuberoot(x))

Homework Equations





The Attempt at a Solution



making a common denominator and expanding:
= lim [tex]\frac{1-3x^{1/3}+2x^{1/2}}{1-x^{1/3}-x^{1/2}+x^{1/6}}[/tex]

I then divided the whole thing by x^1/2 but didnt get anywhere.
Any help would be v much appreciated.
thank you
 
Physics news on Phys.org
  • #2


sara_87 said:
= lim [tex]\frac{1-3x^{1/3}+2x^{1/2}}{1-x^{1/3}-x^{1/2}+x^{1/6}}[/tex]

Simply substituting x=1 into this gives 0/0, so your limit is in one of the forms that qualify for l'hopital's rule...
 
  • #3


ok,thanks so i used L'hospital's rule and got that the limit is 0; is that right?
 
  • #4


Nope, you shouldn't be getting zero...there is actually an error in your first post: [tex]x^{1/3}x^{1/2}=x^{1/3+1/2}=x^{5/6} \neq x^{1/6}[/tex]

You will have to use L'hopistal's rule twice
 
  • #5


you're right
but in that case, when i use l'hospitals rule once, i get 0/-0.5 = 0
so why must i use l'hospital's rule again?
 
  • #6


You should be getting 0/0 after the first time (5/6-1/3-1/2=0 not -0.5)
 
  • #7


Oh my, I am so stupid! i should have known that.
Yep, i see my mistake.
L'hospital's rule twice gives limit =1/2
right?
 
  • #8


Yup! :smile:
 
  • #9


Thanks.
 

1. What does it mean to find the limit of a sequence as n tends to 1?

Finding the limit of a sequence as n tends to 1 means determining the value that the terms of the sequence approach as n gets closer and closer to 1. This value is often denoted by the symbol "L" and is known as the limit of the sequence.

2. How is the limit of a sequence calculated?

The limit of a sequence can be calculated by evaluating the terms of the sequence as n approaches 1. This can be done algebraically or through the use of other methods such as the squeeze theorem or the ratio test.

3. What is the significance of finding the limit of a sequence?

Finding the limit of a sequence helps to determine the behavior and convergence of the sequence. It can also be used to prove the convergence of a series and to find the value of a series.

4. Can the limit of a sequence as n tends to 1 be undefined?

Yes, the limit of a sequence as n tends to 1 can be undefined if the terms of the sequence do not approach a single value as n gets closer to 1. This can happen if the terms of the sequence oscillate or if they approach different values from different directions.

5. Are there any special techniques for finding the limit of a sequence as n tends to 1?

Yes, there are some special techniques that can be used to find the limit of a sequence as n tends to 1. These include the use of L'Hopital's rule, the use of power series, and the use of trigonometric identities. However, these techniques may not always be applicable and the limit may need to be evaluated algebraically or through other methods.

Similar threads

  • Calculus and Beyond Homework Help
Replies
12
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
485
  • Calculus and Beyond Homework Help
Replies
8
Views
767
  • Calculus and Beyond Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
689
  • Calculus and Beyond Homework Help
Replies
4
Views
994
  • Calculus and Beyond Homework Help
Replies
4
Views
867
  • Calculus and Beyond Homework Help
Replies
10
Views
282
  • Calculus and Beyond Homework Help
Replies
22
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
688
Back
Top