https://file.aiquickdraw.com/mathbot/file/1728394960335wo3ygcyc.jpg

I see that the image contains several mathematical limit problems. Let's work through them one by one.

### 1. Problem 1
\[
\lim_{x \to 2} \frac{\sqrt[3]{6 + x} - \sqrt[3]{6 - x}}{x - 2}
\]

This appears to be a limit problem that likely involves indeterminate form \( \frac{0}{0} \) when evaluated directly, and may require the use of L'Hôpital's Rule or a suitable algebraic manipulation.

### 2. Problem 2
\[
\lim_{x \to 0} \frac{x^2 \left( \tan 2x - 2 \tan x \right)^2}{(1 - \cos 2x)^4}
\]

This is a more complex limit, involving trigonometric functions and cosine expansions. A common approach here might be to use small-angle approximations for \( \tan(x) \) and \( \cos(x) \) as \( x \to 0 \).

### 3. Problem 3
\[
\lim_{x \to 0} \frac{\tan^3 x - \sin^3 x}{x^5}
\]

For this one, we can use Taylor series expansions of \( \tan(x) \) and \( \sin(x) \) around \( x = 0 \), as both functions have known expansions, and simplifying them can give the desired result.

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Should I start by working through these problems one by one in more detail?

Let me know if you'd like a solution for a specific one first!

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Here are 5 related questions you might find useful:

1. How does L'Hôpital's Rule apply to indeterminate forms like \( \frac{0}{0} \)?
2. What are the small-angle approximations for \( \sin(x) \) and \( \cos(x) \)?
3. How can Taylor series be used to approximate trigonometric functions around 0?
4. What are the common strategies for handling limits involving powers of trigonometric functions?
5. How can you simplify expressions involving cube roots in limits?

**Tip:** Always check if direct substitution results in an indeterminate form (like \( \frac{0}{0} \)) before proceeding with algebraic manipulations or L'Hôpital's Rule!

Evaluate the following limits:
1. \(\lim_{{x \to 2}} \frac{{\sqrt[3]{{6 + x}} - \sqrt[3]{{6 - x}}}}{{x - 2}}\)
2. \(\lim_{{x \to 0}} \frac{{x^2(\tan 2x - 2 \tan x)^2}}{{(1 - \cos 2x)^4}}\)
3. \(\lim_{{x \to 0}} \frac{{\tan^3 x - \sin^3 x}}{{x^5}}\)

This problem set focuses on evaluating limits using techniques like L'Hôpital's Rule, Taylor series expansions, and small-angle approximations for trigonometric functions. Key concepts include handling indeterminate forms and simplifying expressions with cube roots.

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