Chapter 1: Q26MP (page 45)

Use Maclaurin to do Problems 26 to 29 and check your results by computer.
$\lim_{x \to 0} (\frac{1}{x^{2}} - \frac{1}{1 - \cos^{2} x})$

Short Answer

Expert verified

Using Maclaurin theorem the result is $\lim_{x \to 0} (\frac{1}{x^{2}} - \frac{1}{1 - \cos^{2} x}) = - \frac{1}{3}$ .

Step by step solution

Given information and L Hospital’s rule:

As given:

$\lim_{x \to 0} (\frac{1}{x^{2}} - \frac{1}{1 - \cos^{2} x})$

Formula used: L Hospital’s rule.

If $\lim_{x \to c} \frac{f (x)}{g (x)} = i n d e t e r m i n a t e f r o m {\frac{0}{0}, \frac{\infty}{\infty}, 0 \times \infty}$ , then limit can be found by differentiating top, and bottom functions and applying limit again get $\lim_{x \to c} \frac{f' (x)}{g^{(x)}}$ .

Taylor series:

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

Maclaurin series is the special case of the Taylor series centered at $a = 0$ .

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

Calculation.

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Use Maclaurin to do Problems 26 to 29 and check your results by computer.
$\lim_{x \to 0} (\frac{1}{x^{2}} - \frac{1}{1 - \cos^{2} x})$

Short Answer

Step by step solution

Given information and L Hospital’s rule:

Calculation.

One App. One Place for Learning.

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Use Maclaurin to do Problems 26 to 29 and check your results by computer.limx→0(1x2-11-cos2x)

Short Answer

Step by step solution

Given information and L Hospital’s rule:

Calculation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Force

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Study anywhere. Anytime. Across all devices.

Use Maclaurin to do Problems 26 to 29 and check your results by computer.
$\lim_{x \to 0} (\frac{1}{x^{2}} - \frac{1}{1 - \cos^{2} x})$