Math Problem Statement

Find the limit: lim (x → 0) [(tan(3x²) + sin²(7x)) / x²]

Solution

We are tasked with evaluating the following limit:

limx0tan(3x2)+sin2(7x)x2\lim_{x \to 0} \frac{\tan(3x^2) + \sin^2(7x)}{x^2}

Let's solve this step by step.

Step 1: Apply small-angle approximations

For small xx, we can use the following approximations for tan(x)\tan(x) and sin(x)\sin(x):

  • tan(x)x\tan(x) \approx x when xx is close to 0.
  • sin(x)x\sin(x) \approx x when xx is close to 0.

Thus:

  • tan(3x2)3x2\tan(3x^2) \approx 3x^2 for small xx,
  • sin(7x)7x\sin(7x) \approx 7x, and therefore sin2(7x)(7x)2=49x2\sin^2(7x) \approx (7x)^2 = 49x^2.

Step 2: Substitute these approximations

Substitute these small-angle approximations into the given expression:

tan(3x2)+sin2(7x)3x2+49x2=52x2\tan(3x^2) + \sin^2(7x) \approx 3x^2 + 49x^2 = 52x^2

Thus, the limit becomes:

limx052x2x2\lim_{x \to 0} \frac{52x^2}{x^2}

Step 3: Simplify the expression

Simplifying the above expression:

52x2x2=52\frac{52x^2}{x^2} = 52

Final Answer:

limx0tan(3x2)+sin2(7x)x2=52\lim_{x \to 0} \frac{\tan(3x^2) + \sin^2(7x)}{x^2} = 52

If you have any questions or want more details, feel free to ask!

Related Questions:

  1. How do you apply L'Hôpital's Rule to limits involving indeterminate forms?
  2. What are the common small-angle approximations for trigonometric functions?
  3. How can you determine if a limit results in an indeterminate form?
  4. What happens if the angle in tan(x)\tan(x) or sin(x)\sin(x) isn't small enough to use the approximation?
  5. What other techniques can be used for evaluating limits as x0x \to 0?

Tip:

Always remember that small-angle approximations like sin(x)x\sin(x) \approx x and tan(x)x\tan(x) \approx x are valid when xx is close to 0, and can significantly simplify limit problems.

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Math Problem Analysis

Mathematical Concepts

Limits
Trigonometry
Small-Angle Approximation

Formulas

tan(x) ≈ x for small x
sin(x) ≈ x for small x

Theorems

Small-Angle Approximation Theorem

Suitable Grade Level

Grades 10-12 or college-level calculus