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Chapter 11: Problem 18

$$\text { Limits Evaluate the following limits using Taylor series.}$$ $$\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1-(x / 2)}{4 x^{2}}$$

Short Answer

Expert verified
The limit of the function as \(x\) approaches 0 is \(-\frac{1}{32}\).

Step by step solution

01

Understand the problem

We are given a function, and we need to find the limit of this function as x approaches 0. We will use Taylor series to expand the given function and compute the limit.
02

Expand the square root function using Taylor series

Let's expand the square root function using Taylor series. The general formula for the Taylor series of a function around a point \(a\) is: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^{n}$$ For the square root function, the Taylor series expansion around 0 is: $$\sqrt{1+x} = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}$$ The first few terms of the Taylor series are: $$\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \dots$$
03

Substitute the Taylor series expansion of the square root function into the given function

Now we will substitute the Taylor series expansion of the square root function into the given function: $$\frac{\sqrt{1+x}-1-(x / 2)}{4 x^{2}} = \frac{1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \dots - 1 - (x / 2)}{4 x^{2}}$$ Simplify the expression: $$\frac{- \frac{x^2}{8} + \frac{x^3}{16} - \dots}{4 x^{2}}$$
04

Cancel out the common terms and simplify

Cancel out the common term \(x^2\) from both numerator and denominator, and simplify the remaining expression: $$\frac{- \frac{1}{8} + \frac{x}{16} - \dots}{4}$$
05

Evaluate the limit

As x approaches 0, the remaining terms with \(x\) in the numerator will also approach 0. Thus, we can now evaluate the limit: $$\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1-(x / 2)}{4 x^{2}} = \lim_{x \rightarrow 0} \frac{- \frac{1}{8} + \frac{x}{16}}{4} = -\frac{1}{32}$$ So, the limit of the given function as x approaches 0 is \(-\frac{1}{32}\).

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Most popular questions from this chapter

Find power series representations centered at 0 for the following functions using known power series. $$f(x)=\frac{3}{3+x}$$

By comparing the first four terms, show that the Maclaurin series for \(\cos ^{2} x\) can be found (a) by squaring the Maclaurin series for \(\cos x,\) (b) by using the identity \(\cos ^{2} x=\frac{1+\cos 2 x}{2},\) or \((c)\) by computing the coefficients using the definition.

Representing functions by power series Identify the functions represented by the following power series. $$\sum_{k=2}^{\infty} \frac{x^{k}}{k(k-1)}$$

Let $$f(x)=\sum_{k=0}^{\infty} c_{k} x^{k} \text { and } g(x)=\sum_{k=0}^{\infty} d_{k} x^{k}$$a. Multiply the power series together as if they were polynomials, collecting all terms that are multiples of \(1, x,\) and \(x^{2} .\) Write the first three terms of the product \(f(x) g(x)\) b. Find a general expression for the coefficient of \(x^{n}\) in the product series, for \(n=0,1,2, \ldots\)

Suppose you want to approximate \(\sqrt{72}\) using four terms of a Taylor series. Compare the accuracy of the approximations obtained using the Taylor series for \(\sqrt{x}\) centered at 64 and \(81 .\)
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