We are asked to find the limit of the given expression as $n$ approaches infinity: $$ \lim_{n \to \infty} \frac{n^2}{(n+1)^2} $$

1. Problem Description

We are asked to find the limit of the given expression as $n$ approaches infinity:

$$$$

\lim_{n \to \infty} \frac{n^2}{(n+1)^2}

$$$$

2. Solution Steps

We want to evaluate the limit

$$$$

\lim_{n \to \infty} \frac{n^2}{(n+1)^2}

$$$$

First, we expand the denominator:

$$$$

(n+1)^2 = n^2 + 2n + 1

$$$$

So we can rewrite the expression as:

$$$$

\lim_{n \to \infty} \frac{n^2}{n^2 + 2n + 1}

$$$$

Now, we divide both the numerator and the denominator by the highest power of $n$ in the denominator, which is $n^2$:

$$$$

\lim_{n \to \infty} \frac{n^2/n^2}{(n^2 + 2n + 1)/n^2} = \lim_{n \to \infty} \frac{1}{1 + \frac{2n}{n^2} + \frac{1}{n^2}}

$$$$

Simplify the expression:

$$$$

\lim_{n \to \infty} \frac{1}{1 + \frac{2}{n} + \frac{1}{n^2}}

$$$$

As $n$ approaches infinity, $\frac{2}{n}$ and $\frac{1}{n^2}$ both approach 0:

$$$$

\lim_{n \to \infty} \frac{2}{n} = 0 \\

\lim_{n \to \infty} \frac{1}{n^2} = 0

$$$$

Therefore, the limit becomes:

$$$$

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

$$$$

3. Final Answer

1