1003047704
Level:
C
Find the limit. \[ \lim\limits_{n\to\infty}\frac{1^2+2^2+\dots+n^2}{n^2+7n-3} \]
Hint: \( 1^2+2^2+\cdots +n^2=\frac16 n(n+1)(2n+1) \).
\( \infty \)
\( 0 \)
\( \frac13 \)
\( -\frac13 \)
\( \frac16 \)
Limit of a sequence
1003047703
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\frac{\frac12+\frac14+\dots+\frac1{2^n}}{\frac13+\frac19+\dots+\frac1{3^n}} \).
\( 2 \)
\( \frac23 \)
\( \infty \)
\( 0 \)
\( \frac32 \)
1003047702
Level:
C
Find the limit:
\[ \lim\limits_{n\to\infty}\left(\frac13+\frac19+\dots+\frac1{3^n} \right) \].
\( \frac12 \)
\( \frac13 \)
\( \frac32 \)
\( \infty \)
\( \frac23 \)
1003047701
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\frac{3+9+\dots+3^n}{4+16+\dots+4^n } \).
\( 0 \)
\( \frac32 \)
\( \infty \)
\( 1 \)
\( \frac34 \)
1003047610
Level:
C
Find the limit \( \lim\limits_{n\to\infty} \frac{\sqrt{4n+5}}{\sqrt{2n^3+3n^2-1}} \).
\( 0 \)
\( \sqrt2 \)
\( 2 \)
\( \infty \)
\( -5 \)
1003047609
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\frac{\sqrt{4n^2+3n+1}}{\sqrt{2n^2-5n-7}} \).
\( \sqrt2 \)
\( 2 \)
\( -\frac17 \)
\( 0 \)
\( \infty \)
1003047608
Level:
C
Choose the step to take first to efficiently evaluate the limit of the sequence \( \left( \frac{3n+2}{\sqrt{n^2-1}} \right)_{n=1}^{\infty} \).
We divide the numerator and the denominator by \( n \).
We take \( \sqrt n \) outside brackets in the numerator and the denominator.
We square the denominator.
We divide the numerator by \( n \).
We divide the denominator by \( n \).
1003047607
Level:
C
Find the limit \( \lim\limits_{n\to\infty} n\left( \sqrt n-\sqrt{n-1} \right) \).
\( \infty \)
\( \frac12 \)
\( 0 \)
\( 2 \)
\( -\infty \)
1003047606
Level:
C
The sequence \( \left( \sqrt n \left( \sqrt n-\sqrt{n-1} \right) \right)_{n=1}^{\infty} \) is:
convergent and \( \lim\limits_{n\to\infty} \sqrt n \left( \sqrt n-\sqrt{n-1} \right) =\frac12 \)
convergent and \( \lim\limits_{n\to\infty} \sqrt n \left( \sqrt n-\sqrt{n-1} \right) =0 \)
convergent and \( \lim\limits_{n\to\infty} \sqrt n \left( \sqrt n-\sqrt{n-1} \right) =2 \)
divergent and \( \lim\limits_{n\to\infty} \sqrt n \left( \sqrt n-\sqrt{n-1} \right) =\infty \)
divergent and it does not have an infinite limit
1003047605
Level:
C
Find the limit \( \lim\limits_{n\to\infty} \left( \sqrt n-\sqrt{n-1} \right) \).
\( 0 \)
\( \infty \)
\( -\infty \)
\( -1 \)
\( \sqrt2 \)