Solving Limits Involving Trigonometric Functions Using L'H?pital's Rule

In mathematical analysis, understanding how to solve limits that involve trigonometric functions can be quite challenging. One common method to tackle such problems is by utilizing L'H?pital's Rule. This rule is particularly useful for evaluating limits that result in indeterminate forms, such as (frac{0}{0}) or (frac{infty}{infty}). In this article, we will explore a specific example of a limit that involves a trigonometric function and demonstrate how L'H?pital's Rule can be applied to find its solution.

Example Problem

Consider the limit:

[lim_{x to infty} x sinleft(frac{a}{x}right)]

Applying L'H?pital's Rule

First, let's rewrite the expression using a substitution. Let (u frac{a}{x}). As (x to infty), (u to 0). Therefore, we can express the limit in terms of (u):

[lim_{x to infty} x sinleft(frac{a}{x}right) lim_{u to 0} frac{a sin(u)}{u}]

The limit (frac{a sin(u)}{u}) is in the indeterminate form (frac{0}{0}), which allows us to apply L'H?pital's Rule. According to L'H?pital's Rule, if (lim_{u to 0} frac{f(u)}{g(u)}) is of the form (frac{0}{0}) or (frac{infty}{infty}), then:

[lim_{u to 0} frac{f(u)}{g(u)} lim_{u to 0} frac{f'(u)}{g'(u)}]

Applying this rule to our function, we get:

[lim_{u to 0} frac{a sin(u)}{u} lim_{u to 0} frac{a cos(u)}{1}]

Since (lim_{u to 0} cos(u) 1), the limit simplifies to:

[a cdot 1 a]

Thus, the final answer is:

[lim_{x to infty} x sinleft(frac{a}{x}right) a]

Step-by-Step Solution

Substitution Method

Let's consider the original problem step-by-step:

[x sinleft(frac{a}{x}right) frac{a sinleft(frac{a}{x}right)}{frac{a}{x}} frac{a sin(u)}{u}]

Now, applying L'H?pital's Rule:

[lim_{x to infty} frac{a sinleft(frac{a}{x}right)}{frac{a}{x}} lim_{x to infty} frac{frac{d}{dx} left[a sinleft(frac{a}{x}right)right]}{frac{d}{dx} left[frac{a}{x}right]} lim_{x to infty} frac{a cosleft(frac{a}{x}right) cdot left(-frac{a}{x^2}right)}{-frac{a}{x^2}} a cosleft(frac{a}{x}right) to a cdot 1 a]

Alternative Method Using Direct Substitution

Another approach is to directly substitute (u frac{a}{x}), leading to:

[lim_{x to infty} x sinleft(frac{a}{x}right) lim_{u to 0} frac{a sin(u)}{u}]

Using the standard limit (lim_{u to 0} frac{sin(u)}{u} 1), we get:

[a cdot 1 a]

Conclusion

In both methods, we have shown that the limit [lim_{x to infty} x sinleft(frac{a}{x}right) a]. This example demonstrates the power of L'H?pital's Rule in solving limits involving trigonometric functions. L'H?pital's Rule provides a systematic approach to dealing with indeterminate forms, making it an essential tool in calculus.

Keywords

L'H?pital's Rule, Limits, Trigonometric Functions