Find asymptotes of $(2x)/(x-1)^2$

Question

What are the asymptotes of $$\frac{2x}{(x-1)^2}$$ ? I have problems already on domain.

Answer 1

So $f(x)=\frac{2x}{(x-1)^2}$. Here you horizontal asymptote will be found by taking the lim as x $\rightarrow\infty$, which should be $0$. And your veritval asymptotes will be where the denominator vanishes (i.e when $ (x-1)^2=0$).

Answer 2

Asymptote: $\displaystyle{y = ax + b}$.

$$ \lim_{x \to \infty}\left[{2x \over \left(x - 1\right)^{2}} - ax - b\right] = 0\,, \quad \lim_{x \to \infty} \left\{x\left[{2 \over \left(x - 1\right)^{2}} - a - {b \over x}\right]\right\} =0 $$

$$ \lim_{x \to \infty} \left[{2 \over \left(x - 1\right)^{2}} - a - {b \over x}\right] = 0 \quad\Longrightarrow\quad a = 0 $$

$$ \lim_{x \to \infty} \left[{2x \over \left(x - 1\right)^{2}} - b\right] = 0 \quad\Longrightarrow\quad b = 0 $$

$$ \mbox{Asymptote:}\quad y = 0 $$