Calculate a lim $\lim_{x\to \infty } \left ( \frac{x^2+2}{x^2-4} \right)^{9x^2} $ $$
\lim_{x\to \infty } \left ( \frac{x^2+2}{x^2-4} \right)^{9x^2} 
$$
Can you help with it?

Hint
Let $$A=\left ( \frac{x^2+2}{x^2-4} \right)^{9x^2}$$ and take logarithms of both sides $$\log(A)=9x^2 \log\Big(\frac{x^2+2}{x^2-4}\Big)=9x^2\log\Big(\frac{1+\frac{2}{x^2}}{1-\frac{4}{x^2}}\Big)$$ $$\log(A)=9x^2 \Big[\log \left(1+\frac{2}{x^2}\right)-\log \left(1-\frac{4}{x^2}\right) \Big]$$ Now, use the fact that, for mall $y$, $$\log(1+y)=y-\frac{y^2}{2}+\frac{y^3}{3}+O\left(y^4\right)$$ Apply that to each of the logarithms separately, replace $y$ by its expression as a function of $x$ (that is to say $\frac{2}{x^2}$ for the first and $\frac{-4}{x^2}$ for the second)  and simplify. 
I am sure that you can take from here.
