Showing that $\frac{1+nx^2}{(1+x^2)^n}$ converges to $0$ I aim to show that $\{a_n\} = \frac{1+nx^2}{(1+x^2)^n}$ converges to $0$.  The following two facts seem obvious:
(1) $\forall n \in \mathbb{N}, a_n \ge 0$ (since each $a_n$ is the product of two positive numbers).
(2) $\forall n \in \mathbb{N}, a_n \ge a_{n+1}$.
Yet this is not quite enough to show that $a_n \rightarrow 0$ since the sequence could very well converge to some other $c > 0$.

The claim is not entirely true; it is valid if and only if $ x \in \mathbb{R} \setminus \{ 0 \} $. Here is a solution that uses neither transcendental functions nor calculus techniques such as l'Hôpital's Rule.
Firstly, observe that the limit is $ 1 $ if $ x = 0 $. Next, let $ n \in \mathbb{N} $. By the Binomial Theorem, we have
$$
\forall x \in \mathbb{R}: \quad (1 + x^{2})^{n} \geq 1 + n x^{2} + \frac{n(n - 1)}{2} x^{4}.
$$
Hence,
\begin{align}
\forall x \in \mathbb{R}: \quad \frac{(1 + x^{2})^{n}}{1 + n x^{2}}
&\geq \frac{1 + n x^{2} + \frac{n(n - 1)}{2} x^{4}}{1 + n x^{2}} \\\\\\
&=    \frac{1 + n x^{2}}{1 + n x^{2}} + \frac{\frac{n(n - 1)}{2} x^{4}}{1 + n x^{2}} \\\\\\
&=    1 + \frac{\frac{n(n - 1)}{2} x^{4}}{1 + n x^{2}} \\\\\\
&=    1 + \frac{\frac{1}{n} \left[ \frac{n(n - 1)}{2} x^{4} \right]}{\frac{1}{n} \left( 1 + n x^{2} \right)} \\\\\\
&=    1 + \frac{\frac{n - 1}{2} x^{4}}{\frac{1}{n} + x^{2}} \\\\\\
&=    1 + \frac{n - 1}{2} \cdot \frac{x^{4}}{\frac{1}{n} + x^{2}} \\\\\\
&\geq 1 + \frac{n - 1}{2} \cdot \frac{x^{4}}{1 + x^{2}}.
\end{align}
If $ x \neq 0 $, then the expression in the last line diverges to $ \infty $ as $ n \rightarrow \infty $. It follows that
$$
\forall x \in \mathbb{R} \setminus \{ 0 \}: \quad \lim_{n \rightarrow \infty} \frac{(1 + x^{2})^{n}}{1 + n x^{2}} = \infty.
$$
The original limit must therefore be $ 0 $.
