The problem asks to find the limit of the sequence:

$\lim_{n \to \infty} \tan^{-1} \left( \frac{10n}{10n + 4} \right)$

Let's approach this step by step.

Step 1: Simplify the expression inside the inverse tangent function.

We have:

$\frac{10n}{10n + 4}$

Factor the denominator:

$\frac{10n}{10n + 4} = \frac{10n}{10n(1 + \frac{4}{10n})} = \frac{10n}{10n \left(1 + \frac{4}{10n}\right)} = \frac{1}{1 + \frac{4}{10n}}$

Step 2: Find the limit of the simplified expression as $n \to \infty$.

As $n \to \infty$, the term $\frac{4}{10n} \to 0$, so:

$\lim_{n \to \infty} \frac{1}{1 + \frac{4}{10n}} = \frac{1}{1 + 0} = 1$

Step 3: Take the limit of the arctangent.

Now, we need to find:

$\lim_{n \to \infty} \tan^{-1}(1)$

Since $\tan^{-1}(1) = \frac{\pi}{4}$, the limit of the sequence is:

$\lim_{n \to \infty} \tan^{-1} \left( \frac{10n}{10n + 4} \right) = \frac{\pi}{4}$

Conclusion:

The limit of the sequence is $\frac{\pi}{4}$.

Would you like further details or have any questions about this problem?

Here are five related questions to explore:

1. What is the limit of $\tan^{-1}(x)$ as $x \to \infty$?
2. How does factoring help in simplifying limits of sequences?
3. Can the sequence $\tan^{-1}\left(\frac{10n}{10n + 4}\right)$ ever diverge?
4. What is the behavior of the inverse tangent function for large values of $n$?
5. How does the $\tan^{-1}$ function compare to other inverse trigonometric functions in terms of growth rate?

Tip: When working with limits, especially with trigonometric functions, try to first simplify the expression inside before applying limit properties.