Math Problem Statement

Evaluate the limits involving trigonometric functions, specifically lim x->0 of (7/2) * (sin(3x)/3x) * (1/cos(6x)) and lim x->0 of (tan^3(2x))/(x^2 * sin(3x))

Solution

I see you've uploaded an image with some mathematical expressions and limits. Let's analyze it:

From the top section of the image, I can see a limit expression involving trigonometric functions and the limit as x0x \to 0:

limx072sin(3x)3x1cos(6x)=72\lim_{x \to 0} \frac{7}{2} \cdot \frac{\sin(3x)}{3x} \cdot \frac{1}{\cos(6x)} = \frac{7}{2}

This appears to be using a known limit identity: limx0sin(kx)kx=1\lim_{x \to 0} \frac{\sin(kx)}{kx} = 1 For k=3k = 3, limx0sin(3x)3x=1\lim_{x \to 0} \frac{\sin(3x)}{3x} = 1, and as cos(6x)1\cos(6x) \to 1 when x0x \to 0, the entire expression simplifies to 72\frac{7}{2}.

The second part of the image focuses on: limx0tan3(2x)x2sin(3x)\lim_{x \to 0} \frac{\tan^3(2x)}{x^2 \sin(3x)}

Using approximations for small xx:

  • tan(2x)2x\tan(2x) \approx 2x
  • sin(3x)3x\sin(3x) \approx 3x

Substituting, this becomes: limx0(2x)3x2(3x)=limx08x33x3=83\lim_{x \to 0} \frac{(2x)^3}{x^2 (3x)} = \lim_{x \to 0} \frac{8x^3}{3x^3} = \frac{8}{3}

Thus, the limit simplifies to 83\frac{8}{3}.

Would you like a detailed breakdown of the steps or any clarification on these limits?

Five Related Questions:

  1. What are the standard limit identities for trigonometric functions?
  2. How can L'Hopital's Rule be applied to limits like these?
  3. What is the small angle approximation for sin(x)\sin(x) and tan(x)\tan(x)?
  4. How does the cosine function behave near zero?
  5. What are common trigonometric limits involving powers of xx?

Tip:

When evaluating limits involving trigonometric functions, especially as x0x \to 0, applying small-angle approximations like sin(x)x\sin(x) \approx x and tan(x)x\tan(x) \approx x often simplifies the process significantly.

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometric Functions
Small Angle Approximation

Formulas

lim x->0 (sin(kx)/kx) = 1
sin(x) ≈ x for small x
tan(x) ≈ x for small x

Theorems

Small Angle Approximation Theorem
Trigonometric Limit Theorems

Suitable Grade Level

Undergraduate level - Calculus I or II