$lim_{xto infty}((n-sqrt(n^2+1))/sin(1/n))$

A cura di: Administrator

Il limite presenta a numeratore una forma indeterminata $infty -infty$.

Si ha

$lim_{nrightarrowinfty}frac{n-sqrt{n^2+1}}{sinfrac{1}{n}}=$$lim_{nrightarrowinfty}frac{n-sqrt{n^2big(1+frac{1}{n^2}big)}}{sinfrac{1}{n}}=$$lim_{nrightarrowinfty} frac{n-|n| sqrt{1+frac{1}{n^2}}}{sinfrac{1}{n}}=$$lim_{nrightarrowinfty} frac{ncdotBig(1-sqrt{1+frac{1}{n^2}}Big)}{sinfrac{1}{n}}=$$lim_{nrightarrowinfty} frac{1-sqrt{1+frac{1}{n^2}}}{frac{sinfrac{1}{n}}{frac{1}{n}}}=0$