Limits with the natural logarithm

In summary, the conversation is discussing how to find the limits $\lim_{x\rightarrow 0}\frac{ln(1-x)}{x}$, $\lim_{x\rightarrow 0}\frac{ln(1+x^{2})}{x}$, and $\lim_{x\rightarrow 0}\frac{ln(1+2x)}{x}$ based on the given limit $\lim_{x\rightarrow 0}\frac{ln(1+x)}{x}=1$. The suggested techniques for finding these limits are using change of variables and substitution. The conversation ends with the realization that the limits can be easily computed once an example is seen.
  • #1
Yankel
395
0
Hello

I have three limits to calculate, based on a given limits. What I know is:

\[\lim_{x\rightarrow 0}\frac{ln(1+x)}{x}=1\]

And based on this, I need to find (without L'Hopital rule), the following:

\[\lim_{x\rightarrow 0}\frac{ln(1-x)}{x}\]

\[\lim_{x\rightarrow 0}\frac{ln(1+x^{2})}{x}\]

\[\lim_{x\rightarrow 0}\frac{ln(1+2x)}{x}\]

I can't figure out the technique of moving from the known limits to the ones I need to find.

Thank you in advance !
 
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  • #2
Hi Yankel,

you can try using change of variables for all of them, i.e $u=-x$ for the first one, $u=x^2$ for the second one, etc.
 
  • #3
Ok, so we're given $\lim_{x\to0}\frac{\log(1+x)}{x}=1$ and we have

$$\lim_{x\to0}\frac{\log(1-x)}{x}=\lim_{x\to0}\frac{\log(1+(-x))}{x}=-\lim_{x\to0}\frac{\log(1+(-x))}{-x}$$

Can you compute it now? Can you make progress on the others, both with this method and by using substitution as Rido12 suggested?
 
  • #4
So simple once you see an example...

Thank you !
 

FAQ: Limits with the natural logarithm

What is the natural logarithm?

The natural logarithm, denoted as ln(x), is a mathematical function that gives the logarithm of a number with respect to the base e, where e is a mathematical constant approximately equal to 2.71828.

What is the relationship between limits and the natural logarithm?

Limits with the natural logarithm involve finding the limit of a function that contains the natural logarithm as one of its components. It is a powerful tool in calculus to evaluate complex limits and solve various mathematical problems.

How do you find the limit of a function with the natural logarithm?

To find the limit of a function with the natural logarithm, you can use algebraic manipulation, L'Hospital's rule, or substitution with known limits. It is essential to understand the properties of logarithms and the rules of limits to solve these types of problems.

What are some common applications of limits with the natural logarithm?

Limits with the natural logarithm have various applications in mathematics, physics, and engineering. It is used to find the maximum and minimum values of a function, determine the convergence of series, and solve differential equations, among other things.

Are there any limitations to using limits with the natural logarithm?

Like any other mathematical tool, limits with the natural logarithm have limitations. It may not always be possible to find the limit of a function with the natural logarithm, and the process can be time-consuming and complex. Additionally, it is crucial to check for discontinuities and other special cases when working with limits involving the natural logarithm.

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