Granica ciągu

[bunio244]

\(a) \lim_{n\to \infty } nsin \frac{2}{n}
b) \lim_{n\to \infty } \frac{ \sqrt{n+2}- \sqrt{n+1} }{ \sqrt{n+1}- \sqrt{n} }
c) \lim_{n\to \infty } \frac{1}{ \sqrt{n^2+7n}-n }\)

[radagast]

\(a)
\lim_{n\to \infty } nsin \frac{2}{n}= \left( \frac{2}{n}=t \\n= \frac{2}{t}\\ \lim_{n\to \infty } t=0 \right)=\lim_{t\to 0} \frac{2sint}{t} =2\)

[bunio244]

a można jakoś bez tego t?

[radagast]

\(b)
\lim_{n\to \infty } \frac{ \sqrt{n+2}- \sqrt{n+1} }{ \sqrt{n+1}- \sqrt{n} }=
\lim_{n\to \infty } \frac{ \left( \sqrt{n+2}- \sqrt{n+1} \right)\left( \sqrt{n+2}+ \sqrt{n+1} \right) \left (\sqrt{n+1}+\sqrt{n} \right) }{ \left ( \sqrt{n+1}- \sqrt{n} \right) ( \left( \sqrt{n+2}+ \sqrt{n+1} \right) \left( \sqrt{n+1}+\sqrt{n} \right) }=
\lim_{n\to \infty } \frac{ \left( \sqrt{n+2}- \sqrt{n+1} \right)\left( \sqrt{n+2}+ \sqrt{n+1} \right) \left (\sqrt{n+1}+\sqrt{n} \right) }{ \left( \left (\sqrt{n+1}- \sqrt{n} \right) ( \sqrt{n+2}+ \sqrt{n+1} \right) \left( \sqrt{n+1}+\sqrt{n} \right) }=
\lim_{n\to \infty } \frac{ \left (\sqrt{n+1}+\sqrt{n} \right) }{ \left( \sqrt{n+2}+ \sqrt{n+1} \right) }=
\lim_{n\to \infty } \frac{ \left (\sqrt{1+ \frac{1}{n} }+1\right) }{ \left( \sqrt{1+ \frac{2}{n} }+ \sqrt{1+ \frac{1}{n} } \right) }= \frac{1+1}{1+1}=1\)

[radagast]

a można jakoś bez tego t?

Mażna, np tak:
\(\lim_{n\to \infty } nsin \frac{2}{n}=\lim_{n\to \infty } \frac{sin {\frac{2}{n}}}{ \frac{1}{n} } =2\lim_{n\to \infty } \frac{sin {\frac{2}{n}}}{ \frac{2}{n} } =2\)

[radagast]

\(c) \lim_{n\to \infty } \frac{1}{ \sqrt{n^2+7n}-n }=\lim_{n\to \infty } \frac{ \sqrt{n^2+7n}+n }{ \left( \sqrt{n^2+7n}-n\right) \left( \sqrt{n^2+7n}+n \right) }=\lim_{n\to \infty } \frac{ \sqrt{n^2+7n}+n }{ 7n}=\lim_{n\to \infty } \frac{ \sqrt{1+ \frac{7}{n} }+1}{ 7}= \frac{2}{7}\)