Understanding the Limit of sin(x) / x as x Approaches 0 Using L'H?pital's Rule

The limit of displaystyle lim_{x to 0} frac{sin x}{x} is a classical problem in calculus and one that often arises when studying the behavior of trigonometric functions near zero. This limit evaluates to 1, and various methods can be used to prove this result, including L'H?pital's rule. In this article, we will explore the solution using L'H?pital's rule and provide context on its application in solving indeterminate forms.

Introduction to L'H?pital's Rule

L'H?pital's rule is a powerful technique in calculus used to evaluate limits involving indeterminate forms, such as frac{0}{0} or frac{infty}{infty}. If a limit leads to one of these indeterminate forms, L'H?pital's rule stipulates that the limit of the ratio of two functions is equal to the limit of the ratio of their derivatives, provided these limits exist.

Applying L'H?pital's Rule to the sin(x) / x Limit

Consider the following limit expression:

displaystyle lim_{x to 0} frac{sin x}{x}

This expression represents an indeterminate form of frac{0}{0} since sin(0) 0 and x 0. By L'H?pital's rule:

displaystyle lim_{x to 0} frac{sin x}{x} lim_{x to 0} frac{frac{d}{dx}sin x}{frac{d}{dx}x}

Calculating the derivatives, we get:

displaystyle lim_{x to 0} frac{cos x}{1}

Since the cosine function at 0 is 1, we now have:

displaystyle lim_{x to 0} cos x cos 0 1

Therefore, the limit of frac{sin x}{x} as x approaches 0 is 1.

Other Methods to Evaluate the Limit

While L'H?pital's rule is one straightforward method, this limit can also be evaluated using other approaches such as the sandwich (squeeze) theorem or the Taylor series expansion of the sine function. These methods are mentioned as alternative techniques in many standard calculus textbooks.

The Sandwich Theorem

The sandwich theorem (or squeeze theorem) is another way to prove that displaystyle lim_{x to 0} frac{sin x}{x} 1. The key idea here is to find two functions that both approach the same value at the limit point and that sandwich the given function. In this case, the inequalities |sin x| leq |x| can be used with the squeeze theorem to show that the limit is indeed 1.

Taylor Series Expansion

Another approach involves the Taylor series expansion of the sine function:

sin x x - frac{x^3}{3!} frac{x^5}{5!} - cdots

Dividing by x, we get:

frac{sin x}{x} 1 - frac{x^2}{3!} frac{x^4}{5!} - cdots

As x approaches 0, the higher-order terms vanish, leaving:

frac{sin x}{x} to 1

Again, we see that the limit is 1.

Applications and Importance

The limit of frac{sin x}{x} as x approaches 0 is not just a curiosity in calculus. It plays a significant role in various mathematical and physical contexts. For instance, it is fundamental to the definition of the derivative of the sine function and is used in the analysis of wave behavior in physics.

Conclusion

In conclusion, the limit of frac{sin x}{x} as x approaches 0 is a critical concept in calculus. We have demonstrated its evaluation using L'H?pital's rule, along with other alternative methods. This limit is crucial for understanding the behavior of trigonometric functions and their derivatives, making it a cornerstone in the study of calculus.

Keywords: L'H?pital's rule, Calculus, Indeterminate Forms