Evaluation of $\lim\limits_{n\to\infty} (\sqrt{n^2 + n} - \sqrt[3]{n^3 + n^2}) $

Could you, please, check if I solved it right. \begin{align*} \lim_{n \rightarrow \infty} (\sqrt{n^2 + n} - \sqrt[3]{n^3 + n^2}) &= \lim_{n \rightarrow \infty} \sqrt{n^2(1 + \frac1n)} - \sqrt[3]{n^3(1 + \frac1n)})\\ &= \lim_{n \rightarrow \infty} (\sqrt{n^2} - \sqrt[3]{n^3})\\ &= \lim_{n \rightarrow \infty} (n - n) = 0. \end{align*}

Solution 1:

Here is a very calculus way to do the problem. (Your "taking" $n$ out is useful.) Write $h$ for $1/n$, and rewrite our limit as $$\lim_{h\to 0}\frac{(\sqrt{1+h} -1)-(\sqrt[3]{1+h}-1)}{h}, \tag{1}$$ because $$\frac{\sqrt{1+h} -1}{h} \quad\text{and}\qquad \frac{\sqrt[3]{1+h}-1}{h}$$ are familiar expressions whose limits we can compute. We recognize $$\lim_{h\to 0}\frac{\sqrt{1+h} -1}{h}$$ as the derivative of $\sqrt{1+x}$ at $x=0$, and $$\lim_{h\to 0}\frac{\sqrt[3]{1+h} -1}{h}$$ as the derivative of $\sqrt[3]{1+x}$ at $x=0$. These two derivatives, evaluated at $0$, are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. It follows that the limit (1) is equal to $\dfrac{1}{2}-\dfrac{1}{3}$.

Solution 2:

$$\lim_{n \rightarrow \infty} (\sqrt{n^2 + n} - \sqrt[3]{n^3 + n^2}) $$

$$=\lim_{n \rightarrow \infty} n\left(1+\frac1n\right)^{\frac12}-n\left(1+\frac1n\right)^{\frac13}$$

$$=\lim_{n \rightarrow \infty} n\left(1+\frac1{2n}+O\left(\frac1{n^2}\right)\right)-n\left(1+\frac1{3n}+O\left(\frac1{n^2}\right)\right)$$ (Using Taylor series or Generalized Binomial Theorem)

$$=\lim_{n \rightarrow \infty} \frac12+O\left(\frac1n\right)-\left(\frac13+O\left(\frac1n\right)\right)$$ $$=\frac12-\frac13$$

Alternatively, putting $h=\frac1n,$ $$ (\sqrt{n^2 + n} - \sqrt[3]{n^3 + n^2})=\frac{(1+h)^{\frac12}-(1+h)^{\frac13}}h$$

$$\lim_{h\to0}\frac{(1+h)^{\frac12}-(1+h)^{\frac13}}h $$ can be handled at least in three ways as follows:

$1: $ Using Taylor series,

$$\lim_{h\to0}\frac{(1+h)^{\frac12}-(1+h)^{\frac13}}h=\lim_{h\to0}\frac{1+\frac h2+O(h^2)-(1+\frac h3+O(h^2))}h$$ $$=\frac12-\frac13\text{ as }h\ne0\text{ as }h\to0$$

$2: $ $$\text{As }\lim_{h\to0}\frac{(1+h)^{\frac12}-(1+h)^{\frac13}}h \text{ is of the from }\frac\infty\infty,$$

Applying L'Hospital's Rule, $$\lim_{h\to0}\frac{(1+h)^{\frac12}-(1+h)^{\frac13}}h=\lim_{h\to0}\left(\frac12\cdot(1+h)^{\frac12-1}-\frac1{3}\cdot(1+h)^{\frac13-1}\right)=\frac12-\frac13$$

$3: $ Like Jyrki Lahtonen has approached,

Putting $1+h=y^{\text{lcm}(2,3)}=y^6$

$$\lim_{h\to0}\frac{(1+h)^{\frac12}-(1+h)^{\frac13}}h$$

$$=\lim_{y\to1}\frac{y^3-y^2}{y^6-1}$$

$$=\lim_{y\to1}\frac{y^2(y-1)}{(y-1)(y^5+y^4+y^3+y^2+y+1)}$$

$$=\lim_{y\to1}\frac{y^2}{y^5+y^4+y^3+y^2+y+1}\text{ as }y\ne1\text{ as } y\to1$$

$$= \frac{1^2}{1+1+1+1+1+1}$$

Solution 3:

Others have spotted your error, and described ways of seeing that the limit is $1/6$. If you want to see a non-calculus way, then I suggest splitting it to two parts as follows: $$ \sqrt{n^2+n}-\root3\of{n^3+n^2}=\left(\sqrt{n^2+n}-n\right)-\left(\root3\of{n^3+n^2}-n\right). $$ This operation is valid, if you can show that both sequences on the r.h.s converge. To see that I would interpret $n=\sqrt{n^2}$ in the first summand and $n=\root3\of{n^3}$ in the second. Then use the usual tricks $$ \sqrt a-\sqrt b=\frac{a-b}{\sqrt a+\sqrt b}, $$ and $$ \root 3\of a-\root3\of b=\frac{a-b}{a^{2/3}+(ab)^{1/3}+b^{2/3}}. $$ These come from the respective polynomial factorizations $$ x^2-y^2=(x-y)(x+y),\qquad x^3-y^3=(x-y)(x^2+xy+y^2). $$

Solution 4:

Following your idea:

$$ (\sqrt{n^2 + n} - \sqrt[3]{n^3 + n^2}) = \sqrt{n^2\left(1 + \frac1n\right)} - \sqrt[3]{n^3\left(1 + \frac1n\right)}=$$

$$=n\left(\sqrt{1+\frac1n}-\sqrt[3]{1+\frac1n}\right)=\frac{\sqrt{1+\frac1n}-\sqrt[3]{1+\frac1n}}{\frac1n}$$

Looking at the above as function of the continuous variable $\,n\,$, we can use l'Hospital to try to calculate its limit:

$$\lim_{n\to\infty}\frac{\sqrt{1+\frac1n}-\sqrt[3]{1+\frac1n}}{\frac1n}\stackrel{\text{l'H}}=\lim_{n\to\infty}\left(\frac1{2\sqrt{1+\frac1n}}-\frac1{3\sqrt[3]{\left(1+\frac1n\right)^2}}\right)=\frac12-\frac13=\frac16$$

Solution 5:

You made a mistake when you applied the limit $n \to \infty$ to the $\left(1 + \frac 1n\right)$ terms but not other terms outside. This is not a valid limit application because the whole thing is still in an indeterminate $\infty - \infty$ form.

I can offer you a tedious method that does not require you to estimate the growth of the two terms in the difference. (In fact, if you expand out everything in $O(\ldots)$, you only need basic algebra and some fundamental limit theorems.)

Define \begin{align*} A_n & = \sqrt{n^2 + n} = \sqrt[6]{(n^2 + n)^3} = \sqrt[6]{n^6 + 3n^5 + 3n^4 + n^3} = n\sqrt[6]{1 + O(1/n)}\\ B_n & = \sqrt[3]{n^3 + n^2} = \sqrt[6]{(n^3 + n^2)^2} = \sqrt[6]{n^6 + 2n^5 + n^4} = n\sqrt[6]{1 + O(1/n)}. \end{align*} We want to find $\lim_{n\to\infty} A_n - B_n$. Let us factor $A_n^6 - B_n^6$ as follows: \begin{align*} A_n^6 - B_n^6 & = (A_n - B_n)(A_n^5 + A_n^4B_n + A_n^3B_n^2 + A_n^2B_n^3 + A_nB_n^4 + B_n^5) \\ \therefore A_n - B_n & = \frac{A_n^6 - B_n^6}{A_n^5 + A_n^4B_n + A_n^3B_n^2 + A_n^2B_n^3 + A_nB_n^4 + B_n^5} \end{align*} We know that $A_n^6 - B_n^6 = n^5 + 2n^4 + n^3 = n^5(1 + O(1/n))$. All terms in the denominator are of the form $A^i_nB^{5-i}_n$, so $$ A^i_nB^{5-i}_n = \sqrt[6]{(n^6 + 3n^5 + 3n^4 + n^3)^i(n^6 + 2n^5 + n^4)^{5-i}} = n^5\sqrt[6]{1 + O(1/n)}. $$ Therefore, \begin{align*} \lim_{n\to\infty} A_n - B_n & = \lim_{n\to\infty}\frac{n^5(1 + O(1/n))}{6n^5\sqrt[6]{1 + O(1/n)}} \\ & = \frac 16 \end{align*}