Limit Question: a better way to solve this limit? where $x \to -\infty$

Question

Hey guys so I have this limit:

$$\lim_{x \to -∞} f(x) = {(x+\sqrt{x^2+2x})}$$

I solved it by multiplying numerator and denominator by $$x-\sqrt{x^2+2x}$$ and got $-1$ as my answer, but I really don't like how I solved it; any better way to solve it?

Answer 1

You are looking, setting $t=|x|$, at $$ \lim_{t\to\infty} \sqrt{t^2-2t} - t $$ For the sake of it, here is a solution using the Taylor expansion of $\sqrt{1+u}$ around $0$: for $t>0$, $$ \sqrt{t^2-2t} - t = t\sqrt{1-\frac{2}{t}} - t = t\left(\sqrt{1-\frac{2}{t}} - 1\right) = t\left(1-\frac{1}{t} + o\left(\frac{1}{t}\right) - 1\right) = -1+o(1) $$ showing the limit is indeed $-1$.

Answer 2

You can set $x=-\dfrac1t\enspace(t>0,\,t\to 0)$. The expression rewrites as $$f(x)=-\frac1t+\sqrt{\frac{1\mathstrut{}}{t^2}-\frac2t}=\frac{-1+\sqrt{\strut 1-2t}}t.$$ This is the rate of variation of the function $\sqrt{1-2t}\;$ from $t=0$, so the limit is $$\bigl(\sqrt{1-2t}\bigr)'_{t=0}=\frac1{2\sqrt{1-2t}}\cdot(-2)\Biggm|_{t=0}=-1.$$

Answer 3

For fun

$t=-x$, let $t >2.$

$(t^2-2t+1-1)^{1/2}-t=$

$((t-1)^2-1)^{1/2} -((t-1)^2)^{1/2}-1;$

Then with $m:=(t-1)^2$.

$(m-1)^{1/2}- m^{1/2} -1$.

$y_m:=(m-1)^{1/2}-m^{1/2}.$

$\lim_{m \rightarrow \infty} y_m=0$.

Can you prove it?

Answer 4

I suggest to flip the sign by $y=-x\to \infty$ then

$$\lim_{x \to -∞} f(x) = {x+\sqrt{x^2+2x}}=\lim_{y \to \infty} f(x) = {-y+\sqrt{y^2-2y}}$$

and then

$$\sqrt{y^2-2y}-y=\left(\sqrt{y^2-2y}-y\right)\cdot\frac{\sqrt{y^2-2y}+y}{\sqrt{y^2-2y}+y}$$

Edit As another alternative we can use binomial series $t\to 0, \,(1+t)^n=1+nt+O(t^2)$, are you aware about that?

We proceed as follows

$$\sqrt{y^2-2y}=\sqrt{y^2}\cdot \sqrt{1-2/y}=y\cdot(1-2/y)^\frac12=y\cdot(1-1/y+O(1/y^2))=y-1+O(1/y),$$

then

$$\sqrt{y^2-2y}-y=y-1+O(1/y)-y=-1+O(1/y) \to -1$$

Indeed the $O(1/y)$ notation collects all the terms in absolute value eventually smaller or equal than $c/y$ with $c$ some positive constant then

$$|O(1/y)|\le \frac c y \to 0 \implies O(1/y)\to 0$$