Limit question with Substituting

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In summary, the limit of (x^1000 - 1)/(x - 1) as x approaches 1 is equal to 1000. This can be proven by multiplying out the brackets and using synthetic division or l'Hopital's theorem.
  • #1
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Fine the limit of (x^1000 - 1)/(x - 1) as x approaches 1.

Solution:(x^1000 - 1)/(x - 1) = (x^999 + x^998 + ... + x + 1)(x - 1)/(x - 1)

= (x^999 + x^998 + ... + x + 1)

Substituting x = 1 I get...

Line (1):1 + 11 + 12 + ... + 1999 = 1 + 999 = 1000

Is there any way to prove Line (1): I know it is obvious but I want to prove it mathematically and without obviously counting it on my fingers.
 
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  • #2
I guess I could say 1 + 1(999) = 1000
 
  • #3
If anyone has a better way of solving it please let me know, I want to learn all I can.
 
  • #4
You could use synthetic division.
 
  • #5
It's one of the surprising solutions that the limit as x --> 1 of (x^N -1)/(x-1) = N ... in the limit, the numerator is N times bigger than the denominator.

The question is to prove the expansion - you can demonstrate it simply enough by multiplying out the brackets so it is not clear what you mean when you want a non finger-counting method.
 
  • #6
Miike012 said:
If anyone has a better way of solving it please let me know, I want to learn all I can.

You could use l'Hopital's theorem, if you have that.
 

1. What is a limit question with substituting?

A limit question with substituting is a type of calculus problem that involves finding the limit of a function as a variable approaches a specific value. In this type of problem, the variable is substituted with the specific value and the resulting expression is evaluated to find the limit.

2. How do I solve a limit question with substituting?

To solve a limit question with substituting, first substitute the variable with the specific value and simplify the resulting expression. Then, evaluate the expression to find the limit. If the resulting expression is undefined, the limit does not exist.

3. Can I use any value for the variable in a limit question with substituting?

No, the value for the variable should be chosen carefully to ensure that the resulting expression is well-defined. It is usually best to choose values that are close to the limit value being asked for.

4. What is the purpose of substituting in a limit question?

The purpose of substituting in a limit question is to simplify the expression and make it easier to evaluate. This can help in determining the behavior of the function at a specific point and finding the limit value.

5. Are there any special cases in limit questions with substituting?

Yes, there are some special cases where substituting may not be the most efficient method for solving a limit question. For example, when dealing with indeterminate forms such as 0/0 or infinity/infinity, other techniques such as L'Hopital's rule may be more useful.

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