Understanding and Computing Limits Involving Trigonometric Functions

This article delves into the process of computing limits that involve trigonometric functions, specifically focusing on the use of Taylor expansion and L'H?pital's rule. We will explore the steps and methods to solve such problems, providing you with a comprehensive understanding of these concepts.

Methods for Computing Limits Involving Trigonometric Functions

The evaluation of limits involving trigonometric functions can be complex, but with the right techniques, it can be simplified significantly. In this section, we will discuss the use of Taylor expansion and L'H?pital's rule to find such limits.

Taylor Expansion

Taylor expansion is a powerful tool in calculus that allows us to express functions as infinite series. For instance, the arcsine function, arcsin(x), can be expanded into a series. Specifically, the expansion of arcsin(x) near zero can be written as:

[text{arcsin}(x) x - frac{1}{6}x^3 text{higher order terms}]

Staying at degree three is reasonable due to the presence of the denominator x^3.

Similarly, the arctangent function can be expanded as:

[text{arctan}(x) x - frac{1}{3}x^3 text{higher order terms}]

L'H?pital's Rule

When dealing with indeterminate forms such as 0/0, L'H?pital's rule provides a way to solve them. The rule states that for functions f(x) and g(x), if:

[lim_{x to a} frac{f(x)}{g(x)} frac{0}{0} text{ or } frac{infty}{infty}]

Then:

[lim_{x to a} frac{f(x)}{g(x)} lim_{x to a} frac{f'(x)}{g'(x)}]

provided the limit on the right-hand side exists. In the case of the limit:

[lim_{x to 0} frac{arcsin x - arctan x}{x^3}]

We apply L'H?pital's rule to transform the problem.

Practical Example: Applying L'H?pital's Rule

Let's evaluate the limit:

[lim_{x to 0} frac{arcsin x - arctan x}{x^3}]

First, we use the derivatives of the involved functions:

[frac{d}{dx} (arcsin x) frac{1}{sqrt{1 - x^2}}][frac{d}{dx} (arctan x) frac{1}{1 x^2}]

Applying L'H?pital's rule for the first time, we get:

[lim_{x to 0} frac{frac{1}{sqrt{1 - x^2}} - frac{1}{1 x^2}}{3x^2}]

Since this still results in an indeterminate form, we apply the rule again:

[lim_{x to 0} frac{-frac{x}{(1 - x^2)^{3/2}} frac{2x}{(1 x^2)^2}}{6x}]

Further simplification:

[lim_{x to 0} frac{1}{sqrt{1 - x^2}} - frac{1}{1 x^2} frac{1}{2}]

Example Question: Computing a Trigonometric Limit

Consider the limit:

[lim_{x to 0} frac{sin x - tan x}{x^3}]

By substituting and using L'H?pital's rule, we find:

[lim_{x to 0} frac{sin x cos x - sin x}{x^3 cos x} lim_{x to 0} frac{sin x - sin x cos x}{x^3 cos x}][lim_{x to 0} frac{sin x cos x - sin x}{x^3 cos x} lim_{x to 0} frac{sin x (1 - cos x)}{x^3 cos x}]

Since (sin x approx x) and (1 - cos x approx frac{x^2}{2}) near zero:

[lim_{x to 0} frac{frac{x^3}{2}}{x^3 cos x} lim_{x to 0} frac{1}{2 cos x} frac{1}{2}]

Conclusion

We have discussed the methods and steps involved in computing limits involving trigonometric functions using Taylor expansion and L'H?pital's rule. These techniques are essential tools in calculus for solving complex limit problems.

Related Keywords: Limits, Trigonometric functions, L'H?pital's rule, Taylor expansion