Did I do anything wrong in this limit evaluation?

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In summary, the conversation discusses the evaluation of a limit using L'Hôpital's rule. The original limit is written as ln(2x)/(1/x), which is then simplified using a substitution and L'Hôpital's rule to obtain the correct answer of 0.
  • #1
pylauzier
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Homework Statement



A problem in my book asks the reader to evaluate lim x→0 xln(2x) using L'Hôpital.

Homework Equations



None


The Attempt at a Solution



I think I got it right but I simply wanted to make sure I did no illegal manipulations (I'm self-studying this stuff).

I started by substituting u = 1/x to get an undeterminate form suitable for L'Hôpital. By doing so, lim x→0 becomes lim u→∞

lim x→0 xln(2x) = lim u→∞ ln(2/u) / u

Applying L'Hôpital:

= lim u→∞ (-u)/2u2

= lim u→∞ (-1)/2u

= 0


I know that's the right answer, but did I make any mistakes? Also, was I allowed to switch lim x→0 by lim u→∞? Thanks in advance!
 
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  • #2
Sure, it's right. You didn't even really need a substitution. Write the original limit as ln(2x)/(1/x) which is equally indeterminant. Then l'Hopital gives (1/x)/(-1/x^2)=(-x). That's still 0.
 
  • #3
Dick said:
Sure, it's right. You didn't even really need a substitution. Write the original limit as ln(2x)/(1/x) which is equally indeterminant. Then l'Hopital gives (1/x)/(-1/x^2)=(-x). That's still 0.

Oh right, that sure is simpler. Thanks again!
 

FAQ: Did I do anything wrong in this limit evaluation?

1. Did I follow the proper steps in evaluating this limit?

The proper steps for evaluating a limit depend on the type of limit and the function being evaluated. However, in general, the steps include simplifying the expression, plugging in the limit value, and using algebraic techniques such as factoring or rationalization.

2. How do I know if I made a mistake in my limit evaluation?

If your answer does not match the answer given in the textbook or by a reliable calculator, then you may have made a mistake in your evaluation. It is also important to check for any algebraic errors or incorrect use of limit rules. Checking your work step by step can also help identify any mistakes.

3. Can I use L'Hospital's rule to evaluate this limit?

L'Hospital's rule can only be used when the limit is in an indeterminate form, such as 0/0 or ∞/∞. It cannot be applied to limits that are already in a determinate form, such as 3/4 or 5.

4. Is it possible to have more than one correct answer for a limit evaluation?

In some cases, there may be more than one correct answer for a limit evaluation. This can happen when there are multiple ways to simplify the expression, or when there are multiple possible limit values that result in a valid answer. However, in most cases, there is only one correct answer.

5. How can I improve my limit evaluation skills?

One way to improve your limit evaluation skills is to practice solving a variety of limit problems. You can also review the different types of limits and their corresponding evaluation methods. Seeking help from a tutor or your professor can also be beneficial in understanding any areas where you may be struggling.

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