Can L'Hopital's Rule be applied to limits with multiple zeros?

In summary, the conversation discusses the problem of solving the limit lim[x appr. 0] of (x^x)/((e^x)-1). The participants mention the possibility of using L'Hopital's rule, but are unsure due to the form of the expression. It is also noted that x^(x-1) is indeterminate and goes to infinity. Ultimately, it is determined that the limit goes to 1/0, which is equal to infinity.
  • #1
rman144
35
0
I have been working with a limit for a while now but cannot for the life of me seem to solve it. Any ideas:


lim[x appr. 0] of (x^x)/((e^x)-1)


I've tried turning x^x into e^(x ln(x)), but the root of my problem is that I'm unsure of whether or not I can use L'Hopital's because technically, (0^0)/0 is not necessarily of the form 0/0.
 
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  • #2
e^x - 1 ~ x. Therefore your expression is ~ x^(1+x) -> 0^1 = 0.
 
  • #3
You make a good point, but, assuming that is correct, wouldn't it go to:

x^(x-1) >>> oo
 
  • #4
rman144 said:
You make a good point, but, assuming that is correct, wouldn't it go to:

x^(x-1) >>> oo

That is right.
This is straight forward
x^x->1
exp(x)-1->0
1/0->infinity
 
  • #5
rman144 said:
You make a good point, but, assuming that is correct, wouldn't it go to:

x^(x-1) >>> oo
You're right. I misread your original expression - I missed the / .
 

FAQ: Can L'Hopital's Rule be applied to limits with multiple zeros?

What is a limit with multiple zeros?

A limit with multiple zeros is a mathematical concept that represents the value that a function approaches as its input approaches a certain value, where the function has multiple zeros at that value.

How is a limit with multiple zeros calculated?

A limit with multiple zeros is typically calculated using algebraic techniques such as factoring, simplifying, and rationalizing the function. Graphing the function can also help visualize the behavior of the function near the multiple zeros.

What is the importance of understanding limits with multiple zeros?

Understanding limits with multiple zeros is important in many areas of mathematics and science, as it allows us to analyze the behavior of functions and make accurate predictions about their values. It also helps in finding the derivatives of functions and determining the convergence of series.

Can a limit with multiple zeros have different values depending on the direction of approach?

Yes, a limit with multiple zeros can have different values depending on the direction of approach. This is because the function may behave differently on either side of the multiple zeros, leading to different limits.

Are there any special techniques for evaluating limits with multiple zeros?

Yes, there are special techniques for evaluating limits with multiple zeros, such as the squeeze theorem and L'Hopital's rule. These techniques can be used to simplify the function and find the limit, even if it involves multiple zeros.

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