Do I even need to use L'Hopital's Rule for this

In summary, the limit of (1+x)1/x as x approaches 0 can be evaluated using L'Hopital's rule. The initial attempt of plugging in 0 is not a valid method as 1/0 is undefined. Taking the logarithm and then applying L'Hopital's rule on the result would be the appropriate approach.
  • #1
tjohn101
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0

Homework Statement


Evaluate the limit analytically if necessary using L’Hopital’s rule:
lim x->0 (1+x)1/x


Homework Equations





The Attempt at a Solution


Well, I can get the thing equal to 1 if I just plug in zero, so do I need to use L'hopital's? This whole thing is very confusing...
 
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  • #2
How can you 'plug in 0'?? 1/0 is undefined. Besides the limit isn't 1. I suggest you use l'Hopital.
 
  • #3
To be honest, it's late and I've been doing this stuff for 9 hours straight. It didn't even cross my mind that 1/0 would be undefined.
 
  • #4
Gotcha. So just take the log and use l'Hopital on the result.
 

1. What is L'Hopital's Rule and when is it used?

L'Hopital's Rule is a mathematical tool used to evaluate limits involving indeterminate forms, such as 0/0 or infinity/infinity. It is typically used when other methods, such as direct substitution, do not yield a solution.

2. How do I know if I need to use L'Hopital's Rule?

If you encounter a limit that results in an indeterminate form, such as 0/0 or infinity/infinity, you may need to use L'Hopital's Rule. However, it is important to first check if other methods, such as factoring or simplifying, can be used to evaluate the limit before resorting to L'Hopital's Rule.

3. Can L'Hopital's Rule be used for all types of limits?

No, L'Hopital's Rule can only be used for limits that result in indeterminate forms. It cannot be used for limits that do not have an indeterminate form, such as limits that approach a finite number or limits that approach infinity or negative infinity.

4. Is it necessary to use L'Hopital's Rule in every indeterminate form?

No, L'Hopital's Rule should only be used when other methods are not applicable. In some cases, using algebraic manipulation or other techniques may be simpler and more efficient in evaluating the limit.

5. Are there any limitations or restrictions to using L'Hopital's Rule?

Yes, there are some limitations to using L'Hopital's Rule. It can only be used for limits involving continuous functions, and both the numerator and denominator of the limit must approach 0 or infinity. It also cannot be used for limits involving trigonometric functions or logarithms.

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