# Maclaurin Limit of lnx: \frac{1}{2}

• MHB
• ToastIQ
In summary, the conversation revolved around finding the limit of a function involving fractions and the use of different methods such as combining fractions and Maclaurin expansion. However, L'Hôpital's Rule was not an option as it was not yet taught in the class.
ToastIQ
Hello,

I'm supposed to calculate the limit of this:

$$\displaystyle \lim_{{x}\to{1}}\left(\frac{x}{x-1}-\frac{1}{\ln x}\right)$$

Combining the fractions:
$$\displaystyle \frac{x}{x-1}-\frac{1}{\ln x} = \frac{x\ln x-x+1}{(x-1)\ln x}$$

The substitute
$$\displaystyle u=x-1 \implies x=1+u$$

then gives

$$\displaystyle \frac{(1+u)\ln(1+u)-1-u+1}{(1+u-1)\ln(1+u)}$$

Maclaurin expansion of $$\displaystyle \ln(1+u)$$ :

$$\displaystyle \ln(1+u) = u-\frac{u^2}{2}+u^3B(u)$$

Am I on the right path or am I completely misunderstanding this problem? It looks weird to me when I try to put it all together and I haven't been able to come to the right solution (which is $$\displaystyle \frac{1}{2}$$ ).

Are you required to use a Maclaurin expansion? If not, I would use L'Hôpital's Rule instead.

MarkFL said:
Are you required to use a Maclaurin expansion? If not, I would use L'Hôpital's Rule instead.

Yeah I've seen an example on this problem with L'Hôpital's Rule. But we haven't learned about L'Hôpital's Rule yet, and even if we had, we're not allowed to use it in this class.

## 1. What is the Maclaurin Limit of lnx?

The Maclaurin Limit of lnx is represented by the following equation: limx->0 ln(x) = 0. This means that as x approaches 0, the natural logarithm of x also approaches 0.

## 2. How is the Maclaurin Limit of lnx calculated?

The Maclaurin Limit of lnx can be calculated using the Maclaurin series, which is a special case of the Taylor series. The series is as follows: ln(x) = (x - 1) - (x - 1)^2/2 + (x - 1)^3/3 - (x - 1)^4/4 + ... By plugging in x=0, we can see that the limit is equal to 0.

## 3. Why is the Maclaurin Limit of lnx important?

The Maclaurin Limit of lnx is important because it is a fundamental concept in calculus and is used to calculate derivatives of functions. It is also commonly used in physics and engineering to model natural phenomena.

## 4. Can the Maclaurin Limit of lnx be applied to other functions?

Yes, the Maclaurin Limit of lnx can be applied to other functions as well. It is a general method for finding the limit of a function as the input approaches 0. However, the specific series used to calculate the limit may vary depending on the function.

## 5. Are there any real-world applications of the Maclaurin Limit of lnx?

Yes, the Maclaurin Limit of lnx has many real-world applications. For example, it is used in finance to calculate compound interest, in physics to model the behavior of oscillating systems, and in biology to model population growth. It is also used in statistics and data analysis to fit curves to data points.

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