Question

Calculaţi limita:
lim când x tinde la infinit a lui (x+1/radical din (x^2+2) ) totul la puterea x.

Answer 1

[tex]\displaystyle \lim_{x \to \infty} (\frac{x+1}{ \sqrt{x^2+2} } )^x=[1^{\infty}]=\\ = \lim_{x \to \infty} (1+\frac{x+1}{ \sqrt{x^2+2} }-1 )^x=\\ = \lim_{x \to \infty} (1+\frac{x+1-\sqrt{x^2+2} }{\sqrt{x^2+2} } )^x=\\ =\lim_{x \to \infty} [ (1+\frac{x+1-\sqrt{x^2+2} }{\sqrt{x^2+2} } )^{ \frac{\sqrt{x^2+2}}{x+1-\sqrt{x^2+2}} }]^{x\cdot \frac{x+1-\sqrt{x^2+2} }{\sqrt{x^2+2} }}=\\ =e^{\lim_{x \to \infty \frac{x\cdot(x^2+2x+1-x^2-2)}{\sqrt{x^2+2} (x+1+\sqrt{x^2+2} )} } =e^1=e[/tex]