Solving ∞^0 Indetermination: L'Hospital's Rule Help Needed

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In summary, the conversation discusses finding the limit of (1+2^x)^(1/x) as x approaches infinity, and suggests using L'Hopital's rule. The conversation also mentions using ln to simplify the expression and taking the exponent to find the final answer.
  • #1
alejandro7
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Hi, I'm having troubles wiith this problem:

limit when x->∞ (1+2^x)^(1/x)


I don't know how to proceed (I know I have to use l'Hospital's rule). It's a ∞^0 indetermination.


Thanks!
 
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  • #2
Try letting

[itex]y=ln((1+2^{x})^{\frac{1}{x}})[/itex]

Then what can you do??
 
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  • #3
1/x goes down.
 
  • #4
now what form is it in? Can you use L'Hopitals rule now?
 
  • #5
Don't forget, that now

[itex]e^{y}=(1+2^{x})^{\frac{1}{x}}[/itex]

So when you find y, the limit of

[itex]y=ln((1+2^{x})^{\frac{1}{x}})[/itex]

you have to take [itex]e^{y}[/itex] to get the answer to the limit you're looking for.
 
  • #6
Ok I have:

e^lim when x-> of ((ln(1-2^x)/x))

L'Hôpital now?
 
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  • #7
Well you should have

lim x-->∞ [itex]\frac{ln(1+2^{x})}{x}[/itex]

Then use L'Hopitals rule.

You will find the limit of this. To get the answer you want you have to exponentiate it (since you took the natural log in order to find it).
 

FAQ: Solving ∞^0 Indetermination: L'Hospital's Rule Help Needed

1. What is L'Hospital's Rule?

L'Hospital's Rule is a mathematical theorem used to solve indeterminate forms in calculus. It states that if the limit of a quotient of two functions is in an indeterminate form, then the limit can be evaluated by taking the derivative of the numerator and denominator separately and then finding the limit of the resulting quotient.

2. How do you know when to use L'Hospital's Rule?

L'Hospital's Rule is used when the limit of a quotient of two functions is in an indeterminate form, such as 0/0 or ∞/∞. These indeterminate forms cannot be solved using basic algebraic methods, so L'Hospital's Rule is applied to evaluate the limit.

3. What are some common mistakes when using L'Hospital's Rule?

One common mistake when using L'Hospital's Rule is not checking if the indeterminate form is actually present. Sometimes, the limit can be evaluated using basic algebraic methods instead of using L'Hospital's Rule. It is also important to make sure that both the numerator and denominator of the original function are differentiable.

4. Can L'Hospital's Rule be applied multiple times?

Yes, L'Hospital's Rule can be applied multiple times if the resulting limit is still in an indeterminate form. However, it is important to keep in mind that each time the rule is applied, the quotient becomes more complex and difficult to solve, so it is best to only use it as many times as necessary.

5. Are there any limitations to using L'Hospital's Rule?

Yes, there are some limitations to using L'Hospital's Rule. It can only be applied to limits that are in indeterminate forms, and the functions involved must be differentiable. Additionally, it may not work if the functions involved are not continuous or if the limit is approaching a point of discontinuity.

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