Without using L'Hopital's rule, how can I calaculate this limit?

In summary, the conversation discusses finding the limit of the function (xn-an)/(x-a) without using L'Hopital's rule. The speaker asks for help in getting rid of the indeterminations and suggests factoring x^n-a^n. They also mention the possibility of using l'Hopital's rule if n is not a positive integer. The conversation ends with the speaker realizing that factoring x^n-a^n starting with (x-a)(...) will result in the sum of the other factors adding up to n.a^n-1.
  • #1
Calabi_Yau
35
1
Without using L'Hopital's rule how can I calculate the limit of this function: (xn-an)/(x-a) when x→a

I cannot get rid of the indeterminations no matter what. I would like if you could help me out on this.
 
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  • #2
Calabi_Yau said:
Without using L'Hopital's rule how can I calculate the limit of this function: (xn-an)/(x-a) when x→a

I cannot get rid of the indeterminations no matter what. I would like if you could help me out on this.

Factor x^n-a^n. For example x^2-a^2=(x-a)(x+a). x^3-a^3=(x-a)(x^2+xa+a^2) etc.
 
  • #3
Calabi_Yau said:
Without using L'Hopital's rule how can I calculate the limit of this function: (xn-an)/(x-a) when x→a

I cannot get rid of the indeterminations no matter what. I would like if you could help me out on this.

Is n a positive integer? If so, just factor, as Dick has already suggested. If n is not a positive integer, you more-or-less need to use l'Hosptial's rule, whether you want to or not.
 
  • #4
I figured that out. Took me a while to notice if I factorized x^n-a^n beginning with (x-a)(...) the sum of the other factors would add up to n.a^n-1.

PS: I apologize not having posted my original atempt in solving the problem like the rules require.
 

FAQ: Without using L'Hopital's rule, how can I calaculate this limit?

How do I calculate limits without using L'Hopital's rule?

There are several methods you can use to calculate limits without using L'Hopital's rule. These include using algebraic manipulation, using basic limit properties, and using special limits such as the sandwich theorem.

Why should I avoid using L'Hopital's rule to calculate limits?

While L'Hopital's rule can be a useful tool, it should not be relied on too heavily when calculating limits. This is because it only works for certain types of limits and can sometimes give incorrect results.

Can I use L'Hopital's rule for all types of limits?

No, L'Hopital's rule only works for certain types of limits, specifically in the form of "0/0" or "∞/∞". If a limit is not in this form, then L'Hopital's rule cannot be applied.

What are some common mistakes people make when using L'Hopital's rule?

One common mistake is misapplying L'Hopital's rule by using it when the limit is not in the form of "0/0" or "∞/∞". Another mistake is not simplifying the original limit expression before applying L'Hopital's rule.

Are there any alternatives to L'Hopital's rule for calculating limits?

Yes, there are several alternatives to L'Hopital's rule. These include using algebraic manipulation, using basic limit properties, and using special limits such as the sandwich theorem.

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