Sa se calculeze
[tex]\lim_{x \to \ 0} (x^{201}+x+1)^{\frac{1}{x} }[/tex]

Răspuns:

e

Explicație pas cu pas:

[tex]\lim\limits_{x\to 0}(x^{201}+x+1)^{\dfrac{1}{x}}=\lim\limits_{x\to 0}(1+x^{201}+x)^{\dfrac{1}{x}}= \\ \\ =\lim\limits_{x\to 0}\left[(1+x^{201}+x)^{\dfrac{1}{x^{201}+x}\right]^\dfrac{x^{201}+x}{x}} = e^{\Big{\lim\limits_{x\to 0}\dfrac{x^{201}+x}{x}}} = \\ \\ = e^{\Big{\lim\limits_{x\to 0}(x^{200}+1})}} = e^{1} = \boxed{e}[/tex]