Verification of Limit Using Taylor Expansion: x-ln(1+x)/x^2 = 1/2

In summary, the Taylor expansion method is used to approximate a function using a polynomial of infinite degree. It is derived using the Taylor series and is based on the values and derivatives of the function at a nearby point. The key assumptions for using this method are that the function must be infinitely differentiable and the interval of interest must be small enough for the approximation to be accurate. It can be used to find the value of a function at a point by truncating the Taylor series and the accuracy of the approximation increases with the number of terms included. However, limitations of this method include the requirement of infinitely differentiable functions, potential inaccuracy if the interval is too large or the function has non-linear behavior, and the computational intensity of calculating higher order
  • #1
Fernando Revilla
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I quote a question from Yahoo! Answers

Use series to evaluate lim x->0 (x-ln(1+x)/x^2)?
I got 1/2. Could anyone please verify my answer. I am still very confused about series. Please show how you got your answer.
Thanks

I have given a link to the topic there so the OP can see my response.
 
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  • #2
Certainly, the limit is $1/2:$
$$\lim_{x\to 0}\frac{x-\log (1+x)}{x^2}=\lim_{x\to 0}\frac{x-\left(x-\frac{x^2}{2}+o(x^2)\right)}{x^2}\\
=\lim_{x\to 0}\left(\frac{1}{2}-\frac{o(x^2)}{x^2}\right)= \frac{1}{2}-0=\frac{1}{2}$$
 

FAQ: Verification of Limit Using Taylor Expansion: x-ln(1+x)/x^2 = 1/2

What is the Taylor expansion method used for?

The Taylor expansion method is used to approximate a function using a polynomial of infinite degree. It allows for the estimation of a function's values at a certain point, based on the values and derivatives of the function at a nearby point.

How is the Taylor expansion formula derived?

The Taylor expansion formula is derived using the Taylor series, which is an infinite sum of terms that represent the function's values and derivatives at a certain point. By truncating the series at a certain degree, we can obtain a polynomial that closely approximates the function.

What are the key assumptions for using the Taylor expansion method?

The key assumptions for using the Taylor expansion method are that the function must be infinitely differentiable, meaning that all of its derivatives exist and are continuous. Additionally, the function must be centered at a specific point and the interval of interest must be small enough for the approximation to be accurate.

How can the Taylor expansion method be used to find the value of a function at a point?

The Taylor expansion method can be used to find the value of a function at a point by truncating the Taylor series at a certain degree and evaluating the resulting polynomial at the desired point. The more terms included in the series, the more accurate the approximation will be.

What are some limitations of using the Taylor expansion method?

Some limitations of using the Taylor expansion method include the fact that it can only be used for functions that are infinitely differentiable, which may not be the case for all functions. Additionally, the approximation may not be accurate if the interval chosen is too large or if the function has highly non-linear behavior. Finally, the calculation of higher order derivatives can be computationally intensive.

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