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The problem requires finding the limit:

\[
\lim_{{x \to 0}} \frac{\tan^2(2x)}{5x^2}
\]

To solve this, let's go through it step-by-step.

### Step 1: Rewrite Using a Trigonometric Identity
We know that \(\tan^2(2x) = (\tan(2x))^2\), so the expression can be rewritten as:

\[
\lim_{{x \to 0}} \frac{(\tan(2x))^2}{5x^2}
\]

### Step 2: Apply the Small Angle Approximation
For small angles near zero, \(\tan(2x) \approx 2x\). So, we can approximate \(\tan(2x)\) by \(2x\) when \(x\) is close to zero:

\[
\lim_{{x \to 0}} \frac{(2x)^2}{5x^2} = \lim_{{x \to 0}} \frac{4x^2}{5x^2}
\]

### Step 3: Simplify the Expression
Now, simplify by canceling \(x^2\) in the numerator and denominator:

\[
\lim_{{x \to 0}} \frac{4}{5} = \frac{4}{5}
\]

### Conclusion
The result of the limit is:

\[
\frac{4}{5}
\]

Would you like a deeper explanation of any of these steps? 

Here are 5 related questions to explore:

1. How does the small angle approximation \(\tan(x) \approx x\) help in solving trigonometric limits?
2. What other trigonometric limits commonly use approximations for \(x \to 0\)?
3. How would the result change if we used \(\sin\) or \(\cos\) instead of \(\tan\)?
4. What happens if the power in the denominator is higher or lower than in the numerator?
5. Can we use L'Hôpital's Rule to solve this limit? How would that approach look?

**Tip:** When solving trigonometric limits, always consider small-angle approximations or trigonometric identities to simplify the expressions near zero.

Find the limit of (tan^2(2x)) / (5x^2) as x approaches 0.

Solve the limit of (tan^2(2x)) / (5x^2) as x approaches 0 using the small angle approximation and trigonometric identities. This solution is ideal for high school students learning about limits and trigonometric functions.

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