Math Problem Statement

lim⁡x→0(1+tan⁡3x)\placeholder\lim_{x\rarr0}\left(1+\tan^3x\right)^{^{\placeholder{}}}

lim_{x→0} (1 + tan³x)^(1/x³)

Solution

The expression you provided appears to be part of a limit problem:

lim⁡x→0(1+tan⁡3(x))1x3\lim_{x \to 0} \left(1 + \tan^3(x)\right)^{\frac{1}{x^3}}

Would you like me to solve this problem step by step? If there are additional steps or details you'd like clarified, let me know!

Here are some related questions that might interest you:

  1. How do we apply logarithmic techniques to simplify exponential limits?
  2. What is the Taylor expansion for tan⁡(x)\tan(x) near x=0x = 0?
  3. How does the binomial expansion work for limits like this?
  4. How do L'Hôpital's Rule and substitution simplify such limits?
  5. What are the implications of exponentiation in the context of limits?

Tip: For limits involving exponentials, taking the natural logarithm first often simplifies calculations.

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Math Problem Analysis

Mathematical Concepts

Limits
Exponential Functions
Trigonometry

Formulas

Exponential Limit Formula: lim_{x→0} (1 + f(x))^g(x)
Taylor Expansion of tan(x): tan(x) ≈ x + x³/3 + O(x⁵) for small x

Theorems

Taylor Series Expansion
L'Hôpital's Rule for indeterminate forms
Properties of exponential functions

Suitable Grade Level

Undergraduate Mathematics or Advanced High School (Grade 12)

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