Math Problem Statement

(sin3x)/x

Solution

The expression sin(3x)x\frac{\sin(3x)}{x} is undefined at x=0x = 0 because division by zero is not allowed. However, we can explore the behavior of this function for small values of xx, particularly as xx approaches 0, which can be useful in calculus.

Finding the Limit as xx Approaches 0:

limx0sin(3x)x\lim_{x \to 0} \frac{\sin(3x)}{x}

To evaluate this limit, we can use the standard limit result:

limx0sin(kx)x=k\lim_{x \to 0} \frac{\sin(kx)}{x} = k

In this case, k=3k = 3, so:

limx0sin(3x)x=3\lim_{x \to 0} \frac{\sin(3x)}{x} = 3

Thus, the limit of sin(3x)x\frac{\sin(3x)}{x} as xx approaches 0 is 3.

Would you like to explore further details, or do you have any questions?

Here are 5 related questions:

  1. How would you compute sin(kx)x\frac{\sin(kx)}{x} for different values of kk?
  2. What is the derivative of sin(3x)x\frac{\sin(3x)}{x}?
  3. How does the graph of sin(3x)x\frac{\sin(3x)}{x} behave as xx increases?
  4. Can you expand sin(3x)\sin(3x) using a Taylor series around x=0x = 0?
  5. What happens to sin(3x)x\frac{\sin(3x)}{x} for large values of xx?

Tip: Knowing common limits like limx0sin(x)x=1\lim_{x \to 0} \frac{\sin(x)}{x} = 1 can help in evaluating more complex limits efficiently.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Limits
Calculus

Formulas

lim_{x → 0} (sin(kx)/x) = k

Theorems

Sine Limit Theorem

Suitable Grade Level

Grades 11-12