The asymptote to the graph?

Question

$$f(x)=\frac{x-\sqrt{|x^2+x|}}{x}$$

I found that limit at $-\infty$ is $2$ and limit at $+\infty$ is $0$. The limit to $0$ positive is minus infinity: $\lim_{x\to0^+} \frac{x-\sqrt{|x^2+x|}}{x} = -\infty$.

Answer 1

Hint: Write your term in the form

$$1-\sqrt{\left|1+\frac{1}{x}\right|}$$ this tends to $-\infty$ if $x$ tends to $0^+$

Answer 2

There is a vertical asymptote at the line $x=0$. Like you stated $\lim_{x \to 0^{+}} f(x) \rightarrow -\infty.$