# Doubt about two limits (short)

• Felafel
In summary, the limit of the first expression is equal to 1 and the limit of the second expression can be solved using de l'Hospital's rule, which results in the limit being equal to the derivative of f(a) divided by a times f(a).
Felafel

## Homework Statement

Find the limit of
##1): \displaystyle \lim_{n \to +\infty}(\frac{f(a+\frac{1}{n})}{f(a)})^{\frac{1}{n}}##
##2) \displaystyle \lim_{x \to a} (\frac{f(x)}{f(a)})^{\frac{1}{ln(x)-ln(a)}}(=1^{\infty})##
I am not quite sure if i can solve it the way I did, it has been to easy so there's a trick, I'm afraid

## The Attempt at a Solution

##1) \displaystyle \lim_{n \to +\infty}(\frac{f(a+\frac{1}{n})}{f(a)})^{\frac{1}{n}}##
=##\displaystyle \lim_{n \to +\infty} e^{\frac{1}{n} (log(f(a+\frac{1}{n})-log(f(a)))}##=
=##e^0=1##

##2) \displaystyle \lim_{x \to a} (\frac{f(x)}{f(a)})^{\frac{1}{ln(x)-ln(a)}}(=1^{\infty})##
##\displaystyle \lim_{x \to a} e^{ln(\frac{f(x)}{f(a)}) \frac{1}{ln(x)- ln(a)}}##
##\displaystyle \lim_{x \to a} \frac{ln(f(x))-ln(f(a))}{ln(x)- ln(a)}(=\frac{0}{0})## with de l'Hopital
##\displaystyle \lim_{x \to a} \frac{f'(x)}{xf(x)}=\frac{f'(a)}{af(a)}##

Felafel said:

## Homework Statement

Find the limit of
##1): \displaystyle \lim_{n \to +\infty}(\frac{f(a+\frac{1}{n})}{f(a)})^{\frac{1}{n}}##
##2) \displaystyle \lim_{x \to a} (\frac{f(x)}{f(a)})^{\frac{1}{ln(x)-ln(a)}}(=1^{\infty})##
I am not quite sure if i can solve it the way I did, it has been too easy so there's a trick, I'm afraid

## The Attempt at a Solution

##1) \displaystyle \lim_{n \to +\infty}(\frac{f(a+\frac{1}{n})}{f(a)})^{\frac{1}{n}}##
=##\displaystyle \lim_{n \to +\infty} e^{\frac{1}{n} (log(f(a+\frac{1}{n})-log(f(a)))}##=
=##e^0=1##

##2) \displaystyle \lim_{x \to a} (\frac{f(x)}{f(a)})^{\frac{1}{ln(x)-ln(a)}}(=1^{\infty})##
##\displaystyle \lim_{x \to a} e^{ln(\frac{f(x)}{f(a)}) \frac{1}{ln(x)- ln(a)}}##
##\displaystyle \lim_{x \to a} \frac{ln(f(x))-ln(f(a))}{ln(x)- ln(a)}(=\frac{0}{0})## with de l'Hopital
##\displaystyle \lim_{x \to a} \frac{f'(x)}{xf(x)}=\frac{f'(a)}{af(a)}##

The derivative of ln(x) is 1/x.

You have 1/x in the denominator. That becomes x in the numerator. (You could have checked your result with an appropriate example.)

Felafel said:

## Homework Statement

Find the limit of
##1): \displaystyle \lim_{n \to +\infty}(\frac{f(a+\frac{1}{n})}{f(a)})^{\frac{1}{n}}##
##2) \displaystyle \lim_{x \to a} (\frac{f(x)}{f(a)})^{\frac{1}{ln(x)-ln(a)}}(=1^{\infty})##
I am not quite sure if i can solve it the way I did, it has been to easy so there's a trick, I'm afraid

## The Attempt at a Solution

##1) \displaystyle \lim_{n \to +\infty}(\frac{f(a+\frac{1}{n})}{f(a)})^{\frac{1}{n}}##
=##\displaystyle \lim_{n \to +\infty} e^{\frac{1}{n} (log(f(a+\frac{1}{n})-log(f(a)))}##=
=##e^0=1##

##2) \displaystyle \lim_{x \to a} (\frac{f(x)}{f(a)})^{\frac{1}{ln(x)-ln(a)}}(=1^{\infty})##
##\displaystyle \lim_{x \to a} e^{ln(\frac{f(x)}{f(a)}) \frac{1}{ln(x)- ln(a)}}##
##\displaystyle \lim_{x \to a} \frac{ln(f(x))-ln(f(a))}{ln(x)- ln(a)}(=\frac{0}{0})## with de l'Hopital
##\displaystyle \lim_{x \to a} \frac{f'(x)}{xf(x)}=\frac{f'(a)}{af(a)}##

Note: that should be l'Hospital"---like the place with the emergency ward---not "l'Hopital" (although it is pronounced as "lo-pee-tal", with a silent 's').

Ray Vickson said:
Note: that should be l'Hospital"---like the place with the emergency ward---not "l'Hopital" (although it is pronounced as "lo-pee-tal", with a silent 's').

I tend to see it as l'Hôpital, so maybe they read l'Hôpital and didn't know how to type an ô?

thanks everyone!

## 1. What are limits in scientific research?

Limits in scientific research refer to the boundaries or constraints that exist within a study, which may affect the accuracy, reliability, or generalizability of the results. These can include limitations in the sample size, methodology, instrumentation, or external factors such as funding or time constraints.

## 2. How do I determine the limits of my study?

Determining the limits of a study involves critically evaluating the potential barriers or challenges that may impact the research. This can be done through a thorough literature review, consulting with experts in the field, and being transparent about any limitations in the methodology or data analysis methods.

## 3. How do limits impact the validity of research findings?

Limits can have a significant impact on the validity of research findings as they can introduce bias or error into the study. It is important for researchers to acknowledge and address these limits in their research to ensure the validity and reliability of their results.

## 4. Can limits be beneficial in scientific research?

Yes, limits can be beneficial in scientific research as they can help researchers identify potential weaknesses or areas for improvement in their study. By acknowledging and addressing these limits, researchers can ensure the robustness and accuracy of their findings.

## 5. How can I minimize the impact of limits in my research?

To minimize the impact of limits in research, it is important for scientists to be aware of potential limitations and plan accordingly. This can include using multiple methods or approaches, being transparent about any limitations in the study, and continuously evaluating and addressing potential biases throughout the research process.

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