2010005402
Level:
A
Find the limit.
\[
\lim\limits_{n\rightarrow\infty}\left( \frac7{n^2} +3+\frac{7-4n^2}{3+n^2}\right)
\]
\( -1 \)
\( 0 \)
\( \infty \)
\( -\infty \)
\(1 \)
Limit of a sequence
2010005401
Level:
A
Find the following limit.
\[
\lim _{n\to \infty }\left ( \frac{n+1}
{n - 1} + \frac{n -1}
{n + 1}\right )
\]
\(2\)
\(-1\)
\(0\)
\(1\)
2010005309
Level:
C
Find the limit
\[
\lim\limits_{n\to\infty} \left( \frac{1+2+3+\dots+n}{1-2n^2}\right).
\]
\(- \frac{1}{4}\)
\( 1\)
\( 0\)
\( \infty \)
2010005308
Level:
C
Find the following limit.
\[
\lim\limits_{n\to\infty} \left( \frac{1+2+3+\dots+n}{4n^2-3}\right)
\]
\( \frac{1}{8}\)
\( \frac{1}{4}\)
\( 0\)
\( \infty \)
2010005307
Level:
C
Find the limit.
\[
\lim\limits_{n\to\infty}\frac{3+\frac32+\frac34+\dots+\frac3{2^n}}{2+\frac23+\frac29+\dots+\frac2{3^n}}
\]
\( 2 \)
\( \frac32 \)
\( 0 \)
\( \infty \)
\( \frac23 \)
2010005306
Level:
C
Find the limit.
\[
\lim\limits_{n\to\infty}\left(10^{-1} + 10^{-2} + \dots + 10^{-n} \right)
\]
\( \frac19 \)
\( \frac{1}{10} \)
\( \frac{9}{10} \)
\( 0 \)
\( \infty \)
2010005305
Level:
C
Find the limit.
\[
\lim\limits_{n\to\infty}\frac{\sqrt{6n^3-3n+2}}{\sqrt{2n^3+3n-6}}
\]
\( \sqrt3 \)
\( -\frac13 \)
\( 3 \)
\( 0 \)
\( \infty \)
2010005304
Level:
C
Find the limit of the following sequence.
\[
{\left({\left( \root{n}\of{0.5}+\frac{\root{n}\of{0.5}}
{n} \right)}^{n}\right)}_{
n=1}^{\infty }
\]
Hint: The limit of the sequence \({\left({\left(1 + \frac{1}
{n}\right)}^{n}\right)}_{n=1}^{\infty }\)
is the Euler number \(\mathrm{e}\).
\(\frac12 \mathrm{e}\)
\(\mathrm{e}^{\frac12}\)
\(\frac12 + \mathrm{e} \)
\(\infty \)
2010005303
Level:
C
Find the limit of the following sequence.
\[
{\left(\frac{(n^{2} + 4n + 4)^{n}}
{n^{2n}} \right)}_{n=1}^{\infty }
\]
Hint: The limit of the sequence \({\left({\left(1 + \frac{2}
{n}\right)}^{n}\right)}_{n=1}^{\infty }\)
is \(\mathrm{e}^2\), where \(\mathrm{e}\) is the Euler number.
\(\mathrm{e}^{4}\)
\(\mathrm{e}+4\)
\(4\mathrm{e} \)
\(\infty \)
2010005302
Level:
C
Consider the convergent sequence
\[
(a_{n})_{n=1}^{\infty } = \left (\frac{6n^{2} + 10n - 300}
{2n^{2}} \right )_{n=1}^{\infty }
\]
and its limit \(L\). Find the
maximal difference between \(L\)
and the subsequence \((a_{n})_{n=300}^{\infty }\).
(In other words, find the maximal difference between
\(L\) and the terms of the
sequence starting at \(a_{300}\).)
\(0.015\)
\(0.018\)
\(0.036\)
\(3.015\)