$lim_{xto infty}(sqrt(n^4+n^3+5)-n^2)$

A cura di: Administrator

Il limite si presenta in forma indeterminata $infty – infty$.

Si risolve razionalizzando l’espressione

$lim_{nrightarrowinfty} sqrt{n^4+n^3+5} – n^2 =$$lim_{nrightarrowinfty} frac{big(sqrt{n^4+n^3+5} – n^2big)cdot big(sqrt{n^4+n^3+5} + n^2big)}{sqrt{n^4+n^3+5} + n^2}=$$lim_{nrightarrowinfty} frac{n^4+n^3+5-n^4}{sqrt{n^4+n^3+5} + n^2}=$$lim_{nrightarrowinfty} frac{n^3+5}{sqrt{n^4+n^3+5} + n^2}=$$lim_{nrightarrowinfty} frac{n^3cdotbig(1+frac{5}{n^3}big)}{n^2cdotBigg(sqrt{1+frac{1}{n}+frac{5}{n^4}}+1Bigg)}=$$lim_{nrightarrowinfty} ncdot frac{1+frac{5}{n^3}}{sqrt{1+frac{1}{n}+frac{5}{n^4}}+1}=+infty$