# Evaluating $\lim_{x\to \infty } (x +\sqrt[3]{1-{x^3}} )$

$$\lim_{x\to \infty } (x +\sqrt[3]{1-{x^3}} )$$ What method should I use to evaluate it. I can't use the ${a^3}$-${b^3}$ formula because it is positive. I also tried to separate limits and tried multiplying with $\frac {\sqrt[3]{(1-x^3)^2}}{\sqrt[3]{(1-x^3)^2}}$ , but still didn't get an answer. I got -$\infty$, and everytime I am getting $\infty -\infty$ .

Hint: express: $$x+\sqrt[3]{1-x^3}=x-\sqrt[3]{x^3-1}.$$

HINT: You were near to a solution, but the expression for $a^3+b^3$ should be used. $$a+b=\frac{a^3+b^3}{a^2-ab+b^2},$$ which leads to final limit 0.

• woops I didnt even know there was a formula for $a^3 +b^3$ , thanks Commented Mar 31, 2018 at 16:25
• @EndritShabani You can simply replace $b$ by $-b$ in the known expression. Commented Mar 31, 2018 at 16:35

I usually suggest to make the substitution $x=1/t$, so the limit becomes $$\lim_{t\to0^+}\frac{1+\sqrt[3]{t^3-1}}{t}= \lim_{t\to0^+}\frac{1-\sqrt[3]{1-t^3}}{t}= \lim_{t\to0^+}\frac{1-1+\frac{1}{3}t^3+o(t^3)}{t}$$

Use the identity $$a^3+b^3=(a+b)(a^2-ab+b^2)$$ with $a=x$ and $b=\sqrt[3]{1-x^3}$, to get that $$1=(x+\sqrt[3]{1-x^3})(x^2-x\sqrt[3]{1-x^3}+(1-x^3)^{2/3}).$$ Thus $$\lim_{x\to\infty} (x+\sqrt[3]{1-x^3})=\lim_{x\to\infty}\frac{1}{(x^2-x\sqrt[3]{1-x^3}+(1-x^3)^{2/3})} =\lim_{x\to\infty}\frac{1}{x^2(1-(x^{-3}-1)^{1/3}+(x^{-3}-1)^{2/3})}=0.$$

Hint : make the following change of variable $u = \frac{1}{x^3}$

the limit becomes : $$\lim_{u \to 0} \frac{1}{u^{\frac13}}(1 - (1 - u)^{\frac13})$$

and use the fact that near $0$ you have : $(1-u)^{\frac13} \sim 1-\frac13u$

For $n\gt2$ we have

$$\left(x-{1\over x^2}\right)^3=x^3-3+{3\over x^3}-{1\over x^6}\lt x^3-3+{3\over4}\lt x^3-1$$

Hence, since $x^3-1\lt x^3$ is clear, we have

$$0\lt x-\sqrt[3]{x^3-1}\lt x-\left(x-{1\over x^2}\right)={1\over x^2}\to0\quad\text{as }x\to\infty$$

So by the Squeeze Theorem,

$$\lim_{x\to\infty}\left(x+\sqrt[3]{1-x^3}\right)=0$$

How about without calculus? Does $x+\sqrt[3]{1-x^3}$ ever cross the x axis? \begin{align} x+\sqrt[3]{1-x^3}&=0 \\ x&=-\sqrt[3]{1-x^3} \\ x&=\sqrt[3]{x^3-1} \\ x^3&=x^3-1 \\ 0&=-1 \end{align} It doesn't. It's also defined for all $x$. And one point on the curve $f(x)=x+\sqrt[3]{1-x^3}$ is $f(1)=1\ge0$. Therefore $x+\sqrt[3]{1-x^3} \geq 0 \quad \forall x$. The line $y=0$ serves as the greatest lower bound for $f(x)$

The goal now being to bound the function from the top, let there be some arbitrary value, $a>0$, that we wish to find all $x$ such that $f(x)$ is less than $a$. Firstly we need to know when $f(x)=a$

\begin{align} x+\sqrt[3]{1-x^3}&=a \implies 1-x^3=(a-x)^3=a^3-3a^2x+3ax^2-x^3 \implies \\ 0&=3ax^2-3a^2x+a^3-1 \implies a \neq 0 \quad \land \\ x&=\frac{3a^2 \pm D}{6a} =\frac{a}{2} \pm \frac{D}{6a}\\ &\quad \text{discriminant} \ D=9a^4-4(3a)(a^3-1)=(3a)(4-a^3) \\ &\quad D=0 \implies [a=0] \lor \left[a=\sqrt[3]{4}=2^\frac{2}{3}\right] \\ &\quad \quad [a \neq 0] \implies \\ &\quad \quad [a=\sqrt[3]{4}] \implies x=\frac{\sqrt[3]{4}}{2} \pm 0=2^\frac{-1}{3}\\ &\quad \quad f(2^\frac{-1}{3})=2 \cdot 2^\frac{-1}{3} =2^\frac{2}{3} \ \text{which is possibly} \max{f(x)} \\ &\quad D>0 \implies 0<a<2^\frac{2}{3} \\ &\quad D<0 \implies a \in \lbrace (-\infty, 0) \lor (2^\frac{2}{3}, \infty) \rbrace \\ \end{align}

3 points

1. Seeing that the total range of intersection is the $y$ interval $(0, 2^\frac{2}{3}]$, and how the only place where the line $y=a$ within this range intersects with $f(x)$ only $1$ time is when $x=2^\frac{-1}{3}$ i.e. when $a=2^\frac{2}{3}$, the maximum point of $f(x)$ is $(2^\frac{-1}{3},2^\frac{2}{3})$.
2. On the $x$ interval $(2^\frac{-1}{3}, \infty)$, $f(x)$ can either stay the same value or decrease, since there will be no increasing. Our point $(1,1)$ from earlier shows that $f(x)$ decreases on this interval
3. with no least upper bound, yet a greatest lower bound of $0$, the end behavior of $f(x)$ tends to $0$

Probably easier with calculus