Solving a Homework Problem: Finding the Limit to x-->1+

  • Thread starter Wonderballs
  • Start date
In summary, the Attempt at a Solution has the following:-The professor attempted to solve a problem using the attempt at a solution, which he found on a "practise test"-The problem has an equation in the form (x-1)(x^2+x+1) which the professor is having trouble figuring out-There is no general rule for solving higher order polynomials, but the professor knows how to solve cubics and quartics using a general rule.
  • #1
Wonderballs
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Homework Statement


lim |x-1|/x^3-1
x-->1+



Homework Equations





The Attempt at a Solution



This is from a "practise test" and the prof wrote his solution on it... I'm having a hard time figuring out why (x^3-1) = (x-1)(x^2+x+1) which is what he used to find the limit... I am fine at figureing out limits but I just have no idea how he figures those two equations are equal because when i multiply through i get (x^3-2x^2+2x-1)
 
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  • #2
(x-1)(x^2+x+1)=x^3+x^2+x-x^2-x-1=x^3-1 where the first three terms are obtained by multiplying the first term in the first bracket through with the right bracket, and the last three terms are obtained by multiplying the second term in the first bracket through with the right bracket.
 
  • #3
You're professor is right. Try multiplying it out again. If you get the same (wrong) answer again, show your work, and we'll help show you how to get to the correct answer.

Edit:
Well forget that. Cristo just did it for you.
 
  • #4
Lol, thank you on my paper i have it written down as (x-1)(x^2-x+1)...

But I am still wondering how you get from (x^3-1) TO (x-1)(x^2+x+1)
I have tried to factor by grouping but that got me nowhere...
 
Last edited:
  • #5
I don't remember that equation specifically. I do remember that
[tex]x^n-1 = (x-1)(x^{n-1}+x^{n-2}+\cdots+x+1)[/itex]
Equations of this form come up a lot. It is worth knowing.

If not, you have to factor. In general, yech. There is a general rule for solving cubics. I use it so infrequently I have to look it up every time I use it. There is a general rule for solving quartics, which I never use. There is no general rule for higher order polynomials. On the other hand, I run into things like [itex]1/(1+x)[/itex], [itex]1/(x-1)[/itex], and [itex](x^n-1)/(x-1)[/itex] all the time.
 
  • #6
(x^3-1)=(x^3+x^2-x^2+x-x-1)
=(x^3+x^2-x^2)+(+x-x-1)
=-1(-x^3-x^2+x^2)-1(-x+x)
=-1(i don't know...
 
  • #7
You can spot that x-1 is a factor and then use polynomial division. I don't think I've ever used the cubic formula!
 
  • #8
is it called the binomial theorm?
 
  • #9
There is a fairly well known equation for difference of cubes:

[tex](a^3-b^3)=(a-b)(a^2+ab+b^2)[/tex]

It's fairly easy to prove, once you just multiply everything out. There is also an analogue for addition of cubes which I do not provide for the reader.
 

1. What does "limit to x-->1+" mean?

The notation "x-->1+" indicates that the value of x is approaching 1 from the right side, meaning the values of x are getting closer and closer to 1 as they increase.

2. How do I solve a homework problem involving finding the limit to x-->1+?

To solve a homework problem involving finding the limit to x-->1+, you need to determine the value that x is approaching from the right side. Then, you can use various techniques such as substitution, factoring, or algebraic manipulation to find the limit.

3. What are the common mistakes to avoid when solving a homework problem for the limit to x-->1+?

Some common mistakes to avoid when solving a homework problem for the limit to x-->1+ include forgetting to specify the direction of the limit, using incorrect algebraic techniques, and making errors in substitution or simplification.

4. Are there any shortcuts or tricks to solving a homework problem for the limit to x-->1+?

There are some shortcuts or tricks that can be helpful when solving a homework problem for the limit to x-->1+. One technique is to graph the function and observe the behavior near x=1. Another is to use L'Hôpital's rule, which can simplify the problem by taking the derivative of the numerator and denominator separately.

5. Can I use a calculator to solve a homework problem for the limit to x-->1+?

Most calculators are not designed to solve limit problems, so it is best to use algebraic techniques to solve a homework problem for the limit to x-->1+. However, some calculators may have a feature to calculate limits, so be sure to check your calculator's manual or ask your instructor if it is allowed.

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