<s>[INST] Compute the limit of the sequence of functions of $\lim_{n\to \infty} f_n(x) = \frac{x^2}{x^2+(1-nx)^2}$ Compute the limit of the sequence of functions of 
$$
\lim_{n\to \infty} f_n(x) = \frac{x^2}{x^2+(1-nx)^2}.
$$
Attempt.
$$\lim_{n\to \infty} f_n(x) = \lim_{n\to \infty} \frac{x^2}{x^2+(1-nx)^2} = \lim_{n\to \infty} \frac{x^2}{x^2+n^2x^2-2nx + 1} $$
$$= \lim_{n\to \infty} \frac{x^2}{x^2+\frac{x^2}{n^2}-\frac{2x}{n} + \frac{1}{n}} = \lim_{n\to \infty} \frac{x^2}{x^2} = 1$$
 [/INST] Well $$\lim \frac{x^2}{x^2+n^2x^2-2nx + 1} \ne  \lim \frac{x^2}{x^2+\frac{x^2}{n^2}-\frac{2x}{n} + \frac{1}{n}} $$
Note that there is no need to do any algebraic manipulation here; you can just do it by inspection as $n \to \infty$ we immediately see that $$\lim_{n\to \infty}  \frac{x^2}{x^2+(1-nx)^2}=0$$
as the denominator is tends to infinity the fraction must be zero.
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