Lim x→0 (sin x)/x

Definition

The limit of $$\frac{\sin x}{x}$$ as $$x$$ approaches 0 is a fundamental concept in calculus, often used to evaluate indeterminate forms. This limit is equal to 1, demonstrating the behavior of the sine function near zero and highlighting the relationship between trigonometric functions and their limits. Understanding this limit is crucial when applying L'Hôpital's Rule to resolve limits involving indeterminate forms such as $$\frac{0}{0}$$.

5 Must Know Facts For Your Next Test

1. The limit $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ is fundamental in calculus and often serves as a basis for proving other limits.
2. This limit can be intuitively understood using the unit circle, where the sine of an angle relates to the length of the arc on the circle.
3. The limit demonstrates that as $$x$$ approaches 0, the sine function behaves very similarly to the linear function $$y = x$$.
4. L'Hôpital's Rule can be applied here; by differentiating both numerator and denominator, you can also show that this limit equals 1.
5. This limit is crucial for defining derivatives of trigonometric functions and helps establish the foundation for further studies in calculus.

Review Questions

• How does the limit of $$\frac{\sin x}{x}$$ as $$x$$ approaches 0 illustrate the concept of continuity?
  • The limit of $$\frac{\sin x}{x}$$ as $$x$$ approaches 0 demonstrates continuity because it shows that as $$x$$ gets closer to 0, the value of the function approaches a specific value, which is 1. This continuous behavior indicates that there are no breaks or jumps in the function's graph around that point. This property is essential for functions that are differentiable and emphasizes the smooth transition of values in calculus.
• Explain how L'Hôpital's Rule can be applied to evaluate the limit $$\lim_{x \to 0} \frac{\sin x}{x}$$.
  • To apply L'Hôpital's Rule to the limit $$\lim_{x \to 0} \frac{\sin x}{x}$$, you first recognize it is an indeterminate form of type $$\frac{0}{0}$$. By taking the derivative of the numerator, which is $$\cos x$$, and the derivative of the denominator, which is 1, you can rewrite the limit as $$\lim_{x \to 0} \cos x = 1$$. This confirms that the original limit equals 1, demonstrating how L'Hôpital's Rule helps resolve such indeterminate forms efficiently.
• Evaluate and discuss how understanding $$\lim_{x \to 0} \frac{\sin x}{x}$$ impacts other areas in calculus.
  • $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ has significant implications in calculus, particularly in defining derivatives of trigonometric functions. It provides a foundational understanding that leads to evaluating limits involving sinusoids and demonstrates how these functions behave near critical points. This knowledge is applied in various contexts like series expansions through Taylor Series, ultimately enhancing our comprehension of periodic functions and their derivatives, making it an essential part of a student's toolkit in calculus.

Related terms

Indeterminate Form: A mathematical expression that does not have a well-defined limit, often seen in calculus with forms like $$\frac{0}{0}$$ or $$\infty - \infty$$.

L'Hôpital's Rule:

A method in calculus used to evaluate limits of indeterminate forms by differentiating the numerator and denominator.

Taylor Series: An infinite series that represents a function as a sum of terms calculated from the values of its derivatives at a single point.