<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:

  
*
  
*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] You should be aware of Taylor expansion. Substitute $x=1/t$, so $t\to0^+$ and the limit can be rewritten as
$$
\lim_{t\to0^+}\frac{\sqrt{1+t}-\sqrt[4]{1+t+t^2}}{t^{A+1/2}}
$$
The numerator can be rewritten as
$$
1+\frac{1}{2}t-1-\frac{1}{4}{t}+o(t)
$$
So the limit is $1/2$ when $A+1/2=1$. What if $A>-1/2$ or $A<-1/2$?
The second limit can be computed similarly.
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