Math Problem Statement

Evaluate the limit: lim (x -> 0) sin(x)/x

Solution

The image you uploaded shows a limit expression for a trigonometric function:

limx0sinxx\lim_{x \to 0} \frac{\sin x}{x}

This is a well-known limit that evaluates to 1. The limit:

limx0sinxx=1\lim_{x \to 0} \frac{\sin x}{x} = 1

This result is important in calculus and can be derived using L'Hopital's Rule, series expansion, or geometric reasoning.

Would you like more details on how to derive this result, or any specific explanation related to this? Let me know!

Here are a few questions you could explore:

  1. How can L'Hopital's Rule be used to evaluate this limit?
  2. What is the significance of the result limx0sinxx=1\lim_{x \to 0} \frac{\sin x}{x} = 1 in calculus?
  3. How does the small angle approximation of sinx\sin x help in evaluating the limit?
  4. What are other trigonometric limits that are commonly used in calculus?
  5. How can we use this limit to evaluate limits of other trigonometric functions?

Tip: Whenever you encounter indeterminate forms like 00\frac{0}{0}, L'Hopital's Rule or series expansion can often simplify the problem.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Calculus
Limits

Formulas

lim (x -> 0) sin(x)/x = 1

Theorems

L'Hopital's Rule
Small Angle Approximation

Suitable Grade Level

Grades 11-12