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.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Magnetism

Physical Quantities And Units

Nuclear Physics

Further Mechanics and Thermal Physics

Thermodynamics

Oscillations

Study anywhere. Anytime. Across all devices.

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

Magnetism

Physical Quantities And Units

Nuclear Physics

Further Mechanics and Thermal Physics

Thermodynamics

Oscillations

Study anywhere. Anytime. Across all devices.

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