We are tasked with finding the limit of the following expression as $x$ approaches 0:

$\left( \frac{\tan(x)}{x} \right)^{\frac{1}{x^3}}$

Step 1: Analyzing the indeterminate form

We start by substituting $x = 0$ into the expression:

$\left( \frac{\tan(0)}{0} \right)^{\frac{1}{0^3}}$

Since $\tan(0) = 0$ and the expression involves division by 0, this clearly leads to an indeterminate form of the type $0^0$.

Step 2: Taking the natural logarithm

To handle this indeterminate form, we take the natural logarithm of the entire expression and solve the limit of the logarithm first. Let:

$L = \lim_{x \to 0} \left( \frac{\tan(x)}{x} \right)^{\frac{1}{x^3}}$

Then, taking the logarithm of both sides:

$\ln L = \lim_{x \to 0} \frac{1}{x^3} \ln \left( \frac{\tan(x)}{x} \right)$

Step 3: Simplifying the expression

Next, we simplify the term inside the logarithm. We know that as $x \to 0$:

$\frac{\tan(x)}{x} \to 1$

Thus, $\ln \left( \frac{\tan(x)}{x} \right) \to \ln(1) = 0$. However, we need to explore this more carefully because we have the term $\frac{1}{x^3}$ multiplying it. To better understand the behavior of $\ln \left( \frac{\tan(x)}{x} \right)$, we use the approximation for $\tan(x)$ near 0:

$\tan(x) \approx x + \frac{x^3}{3} + O(x^5)$

So,

$\frac{\tan(x)}{x} \approx 1 + \frac{x^2}{3}$

Now, applying the logarithmic approximation for small $x$:

$\ln \left( 1 + \frac{x^2}{3} \right) \approx \frac{x^2}{3}$

Step 4: Substituting the approximation

Substitute this approximation into the limit:

$\ln L = \lim_{x \to 0} \frac{1}{x^3} \cdot \frac{x^2}{3} = \lim_{x \to 0} \frac{x^2}{3x^3} = \lim_{x \to 0} \frac{1}{3x}$

As $x \to 0$, $\frac{1}{3x} \to \infty$. Therefore:

$\ln L \to -\infty$

Step 5: Final conclusion

Since $\ln L \to -\infty$, it follows that:

$L = e^{-\infty} = 0$

Thus, the original limit is:

$\lim_{x \to 0} \left( \frac{\tan(x)}{x} \right)^{\frac{1}{x^3}} = 0$

Conclusion:

The limit of $\left( \frac{\tan(x)}{x} \right)^{\frac{1}{x^3}}$ as $x$ approaches 0 is $0$.

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Do you need further details or explanations on any step? Here are some related questions you could explore:

1. How does the approximation $\tan(x) \approx x + \frac{x^3}{3}$ work for small $x$?
2. What other indeterminate forms can arise in calculus?
3. How can L'Hopital's Rule help in solving limits involving indeterminate forms?
4. What is the significance of the natural logarithm in handling limits of powers?
5. Can you generalize this result for other trigonometric functions?

Tip: Always check the indeterminate form type before applying logarithmic techniques.