Infinite Limit Approaches

Question

I was doing my book-work when I encountered a problem that I did not know how to approach or solve. The problem is as follows. $$\displaystyle \lim_{x \to -\infty} \frac{x}{x-\sqrt{x^2+5}}$$ I did not know which method to use to solve it, and apparently, the answer is $0.5$.

At this point, I have not learned the lhopital rule and I am not supposed to solve it graphically.

Answer 1

$$\begin{array}{rcl} \displaystyle \lim_{x \to -\infty} \frac{x}{x-\sqrt{x^2+5}} &=& \displaystyle \lim_{u \to \infty} \frac{-u}{-u-\sqrt{(-u)^2+5}} \\ &=& \displaystyle \lim_{u \to \infty} \frac{u}{u+\sqrt{u^2+5}} \\ &=& \displaystyle \lim_{u \to \infty} \frac{1}{1+\sqrt{1+5u^{-2}}} \\ &=& \displaystyle \frac{1}{1+\sqrt{1+5\cdot0}} \\ &=& \displaystyle \frac{1}{2} \\ \end{array}$$

Desmos:

Answer 2

$$\lim_{x\rightarrow-\infty}\frac{x}{x-\sqrt{x^2+5}}=\lim_{x\rightarrow-\infty}\frac{1}{1+\sqrt{1+\frac{5}{x^2}}}=\frac{1}{2}.$$

Answer 3

About $\infty$, $x^2+k\sim x^2$ so $$\lim_{x \to -\infty} \frac{x}{x-\sqrt{x^2+5}}=\lim_{x \to -\infty} \frac{x}{x-\sqrt{x^2}}=\lim_{x \to -\infty} \frac{x}{x-|x|}=\lim_{x \to -\infty} \frac{x}{x+x}=\dfrac12$$