<s>[INST] Getting the wrong result when solving a limit I correctly solved to following limit like this:
$$\lim_{x\to\infty} \left(\frac{1}{x^2}\right)^{\frac{2x}{x+1}}$$
$$ = \lim_{x\to\infty} (\frac{1}{x^2})^{\frac{2x}{x(1+\frac{1}{x})}}$$
$$ = \lim_{x\to\infty} (\frac{1}{x^2})^{\frac{2}{(1+\frac{1}{x})}}$$
$$ = 0^2 = 0$$
However, I usually use the following formula for these kind of limits: $\lim_{x\to\infty} (1+\frac{1}{v})^v = e$ This is simple one that doesn't really need this extra complexity, but anyway. The problem is that I get a different and wrong result:
$$\lim_{x\to\infty} (\frac{1}{x^2})^{\frac{2x}{x+1}}$$
$$= \lim_{x\to\infty} (1 + \frac{1}{x^2} - 1)^{\frac{2x}{x+1}}$$
$$= \lim_{x\to\infty} (1 + \frac{1-x^2}{x^2})^{{\frac{2x}{x+1} \frac{1-x^2}{x^2} \frac{x^2}{1-x^2}}}$$
$$= e^{ \lim_{x\to\infty} {{\frac{2x}{x+1} \frac{1-x^2}{x^2}}}}$$
$$= e^{ \lim_{x\to\infty} {{\frac{2x-2x^3}{x^3+x^2}}}} = e^{ \lim_{x\to\infty} {{\frac{-2x^3}{x^3}}}} = e^{-2}$$
 [/INST] $$\frac{1-x^2}{x^2}$$ does not approach zero as $x$ approaches infinity, so you cannot simply write that $$\lim_{x\rightarrow\infty}\left(1+\frac{1-x^2}{x^2}\right)^{\frac{x^2}{1-x^2}}=e$$
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