<s>[INST] Finding limit function $\lim_{n \rightarrow \infty} n ((x^2 +x + 1)^{1/n} -1)$ \begin{align}
f(x) &= \lim_{n \rightarrow \infty} n ((x^2 +x + 1)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n ((\infty)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n (1 -1)\\& = 
 \lim_{n \rightarrow \infty} n \cdot 0 \\&= 0 
\end{align}
Did I solve it correctly??
 [/INST] Note that $$\lim\limits_{n \rightarrow \infty} n ((x^2 +x + 1)^{\tfrac{1}{n}} -1) =\lim\limits_{n\rightarrow \infty}  \dfrac{(x^2 +x + 1)^{\tfrac{1}{n}} -1}{\dfrac{1}{n}}. $$
Denoting $h(x)=x^2 +x + 1,$ we have
$$\lim\limits_{n\rightarrow \infty}  \dfrac{(x^2 +x + 1)^{\tfrac{1}{n}} -1}{\dfrac{1}{n}}=\lim\limits_{n\rightarrow \infty}  \dfrac{(h(x))^{\tfrac{1}{n}} -1}{\dfrac{1}{n}}=\ln{h(x)}.$$
</s>