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The given limit problem is:

\[
\lim_{{n \to \infty}} n \left( \sqrt{25 + \frac{1}{n}} - 5 \right)
\]

Let's solve it step by step:

### Step 1: Simplifying the square root
We need to find the limit as \( n \to \infty \). To do this, let's focus on the term \( \sqrt{25 + \frac{1}{n}} \).

As \( n \to \infty \), \( \frac{1}{n} \) becomes very small, so we can approximate the expression using a binomial expansion or Taylor expansion around \( \frac{1}{n} \).

We have:

\[
\sqrt{25 + \frac{1}{n}} = 5 \sqrt{1 + \frac{1}{25n}}
\]

Using the approximation \( \sqrt{1 + x} \approx 1 + \frac{x}{2} \) for small \( x \), we get:

\[
\sqrt{1 + \frac{1}{25n}} \approx 1 + \frac{1}{50n}
\]

Thus,

\[
\sqrt{25 + \frac{1}{n}} \approx 5 + \frac{1}{10n}
\]

### Step 2: Substituting the approximation into the limit
Now substitute this back into the limit expression:

\[
n \left( \sqrt{25 + \frac{1}{n}} - 5 \right) \approx n \left( \left( 5 + \frac{1}{10n} \right) - 5 \right)
\]

Simplifying the expression:

\[
n \left( 5 + \frac{1}{10n} - 5 \right) = n \left( \frac{1}{10n} \right) = \frac{n}{10n} = \frac{1}{10}
\]

### Step 3: Taking the limit
As \( n \to \infty \), the expression simplifies to:

\[
\lim_{{n \to \infty}} \frac{1}{10} = \frac{1}{10}
\]

Thus, the value of the limit is:

\[
\boxed{\frac{1}{10}}
\]

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:

1. What is the Taylor expansion of \( \sqrt{1 + x} \) for higher-order terms?
2. How can binomial approximations be used for limits involving square roots?
3. How would the limit change if the expression inside the square root were different?
4. What happens if we try to solve this limit without using approximation techniques?
5. What are some common methods for dealing with limits as \( n \to \infty \)?

**Tip**: When dealing with square roots in limits, using approximations like the binomial or Taylor expansion can simplify the process greatly.

Calculate the following \( \lim_{{n \to \infty}} n \left( \sqrt{25 + \frac{1}{n}} - 5 \right) \)

Step-by-step solution to the limit calculation \( \lim_{{n \to \infty}} n \left( \sqrt{25 + \frac{1}{n}} - 5 \right) \), using approximations such as the binomial and Taylor expansion. Suitable for Grades 11-12 or first-year calculus students.

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