<s>[INST] how to solve this two limit tasks Hello i stumbled across this two limits task and i cant find an answer to them:

  
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*Find the limit depending on the parameter $A$
$$\lim \limits_{x\to\infty}\left(\left(\sqrt{x+1} - \sqrt[4]{x^2 + x + 1} \right) \cdot x^A\right)$$

I tried by multipliying with
$$\frac{\sqrt{x+1} + \sqrt[4]{x^2 + x + 1}}{\sqrt{x+1} + \sqrt[4]{x^2 + x + 1}}$$
which equals 1 and then
$$\frac{x+1 + \sqrt{x^2 + x + 1}}{x+1 + \sqrt{x^2 + x + 1}}$$ so i can get rid off roots in denominators but i got tangled up, 


  
*$$\lim\limits_{x\to\infty}\left((x+1) - \sqrt[3]{x^3 + x^2} \right)$$

lim in both tasks goes to +infinity
 [/INST] For 2 we have $\lim\limits_{x\to\infty}\left((x+1) - \sqrt[3]{x^3 + x^2} \right)=\lim\limits_{x\to\infty}\frac{2x^2+3x+1}{(x+1)^2+(x+1)\sqrt[3]{x^3 + x^2} +\sqrt[3]{(x^3 + x^2)^2}}$.
Now you can factor $x^2$ both in the numerator and the denominator and see that the limit is $\frac{2}{3}$.The first limit can be done similarly.
Note: What I used was that $a^3-b^3=(a-b)(a^2+ab+b^2)$,$\forall a,b \in \mathbb{R}$.
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