Fórum de Matemática
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17. Calcule os limites das sequências
a) \(\frac{n}{\left (n ^{2} \right+2014 )}\)
b) \(\frac{\left ( n+1 \right )^{\frac{1}{2}}}{\left (n+2 \right )^{\frac{1}{2}}}\)

a) \(\lim \frac{n}{n^{2}2014}=\lim \frac{1}{\frac{n^{2}2014}{n}}=\lim \frac{1}{n2014}=\frac{1}{+\infty }=0\)

b) Cálculo auxiliar:

\(\frac{n+1}{n+2}=1+\frac{-1}{n+2}\)

\(\lim \frac{\left ( n+1 \right )^{\frac{1}{2}}}{\left ( n+2 \right )^{\frac{1}{2}}}=\lim \left ( \frac{n+1}{n+2} \right )^{\frac{1}{2}}=\lim \left ( 1+\frac{-1}{n+2} \right )^{\frac{1}{2}}=\left (1+\frac{-1}{+\infty } \right )^{\frac{1}{2}}=\left ( 1+0 \right )^{\frac{1}{2}}=1\)

ou a partir do 2º passo \(\left [ \lim \left ( \frac{n+1}{n+2} \right ) \right ]^{\frac{1}{2}}=\left [ \lim \left ( \frac{n}{n} \right ) \right ]^{\frac{1}{2}}=1^{\frac{1}{2}}=1\)


Caso tenha dúvidas não hesite em dizer.