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Chapter 1: Q27P (page 45)

Use Maclaurin to do problem 26 to 29 and check your results by computer.
$\underset{x \to 0}{l i m} (\frac{1}{x^{2}} - c o t^{2} x)$

Short Answer

Expert verified

Answer using Maclaurin theorem is

$\lim_{x \to 0} (\frac{1}{x^{2}} - \cot^{2} x) = \frac{2}{3}$

Step by step solution

01

Given information and L Hospital’s rule:

Given: $\lim_{x \to 0} (\frac{1}{x^{2}} - \cot^{2} x)$

Formula: L Hospital’s rule.

Ifrole="math" localid="1658896015169" $\underset{x \to c}{l i m} \frac{f (x)}{g (x)} = i n d e t e r m i n a t e f o r m {\frac{0}{0}, \frac{\approx}{\infty}, 0 \times \infty}$ , then limit can be found by differentiating top , bottom functions and applying limit again to get $\lim_{x \to c} \frac{f (x)}{g (x)}$

Taylor series:

role="math" localid="1658896326667" $f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (a)}{n!} (x - a)^{n} f (a) + f^{'} (a) (x - a) + \frac{f " (a)}{2!} (x - a)^{2} + \frac{f''' (a)}{3!} (x - a)^{3} + \dots$

Maclaurin series is the special case of the Taylor series centered at :

$f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n} = f (0) + f^{'} (0) x + \frac{f^{'}' (0)}{2!} x^{2} + \frac{f''' (0)}{3!} x^{3} + \dots$

02

Calculation.

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Most popular questions from this chapter

$\frac{s i n (\sqrt{x})}{\sqrt{x}}, x > 0$

Consider the series in Problem 4.6and show that the remainder after n terms is $R_{n} = \frac{1}{n + 1}$ . Compare the value of term $n + 1$ with $R_{n}$ for n=3, n=10, n=100, n=500to see that the first neglected term is not a useful estimate of the Error.

$\sum_{n = 1}^{\infty} {(- 1)}^{n} n^{3} x^{n}$

$\sum_{n = 1}^{\infty} \frac{(- 1)^{n} x^{n}}{(2 n)!}$

$f (x) = s i n x, a = π / 2$

See all solutions

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