1003047404
Level:
B
Find the limit \( \lim\limits_{n\rightarrow\infty}\frac{5^{n+1}+6^n}{5^n+6^{n+1} } \).
\( \frac16 \)
\( \frac56 \)
\( 1 \)
\( 0 \)
\( \infty \)
Limit of a sequence
1003047405
Level:
B
The sequence \( \left(\frac{3^n-4^{n-1}}{4^n}\right)_{n=1}^{\infty} \) is:
convergent and \(\lim\limits_{n\rightarrow\infty}\frac{3^n-4^{n-1}}{4^n}=-\frac14 \)
convergent and \(\lim\limits_{n\rightarrow\infty}\frac{3^n-4^{n-1}}{4^n}=\frac14 \)
convergent and \(\lim\limits_{n\rightarrow\infty}\frac{3^n-4^{n-1}}{4^n}=-1 \)
convergent and \(\lim\limits_{n\rightarrow\infty}\frac{3^n-4^{n-1}}{4^n}=0 \)
divergent
1003047406
Level:
B
Choose the correct formula to calculate the limit. \[ L=\lim\limits_{n\rightarrow\infty}\frac{3^{n+1}+4^n}{2^n} \]
\( L=\lim\limits_{n\rightarrow\infty}\left(3\cdot\left(\frac32\right)^n+2^n\right)=\infty \)
\( L=\lim\limits_{n\rightarrow\infty}\frac{2^n\left(3\cdot\left(\frac32\right)^n+2^n \right)}{2^n}=0 \)
\( L=\lim\limits_{n\rightarrow\infty}\frac{3^n \left(3+\left(\frac43\right)^n\right)}{2^n}=0 \)
\( L=\lim\limits_{n\rightarrow\infty}\frac{7^{n+1}}{2^n}=\infty \)
\( L=\frac{3^{\infty+1}+4^{\infty}}{2^{\infty}} =\frac72 \)
1003047407
Level:
B
Find the limit \( \lim\limits_{n\rightarrow\infty}\frac{3^{n+1}+4^{n+1}}{3^{n-1}+4^{n-1}}\).
\( 16 \)
\( \frac1{16} \)
\( 0 \)
\( \infty \)
\( 1 \)
1003047408
Level:
B
Choose the best first step to simplify and calculate the limit \(\lim\limits_{n\rightarrow\infty}\frac{3^n+4^{n-1}}{3^n+4^{n+1}} \).
We divide the numerator and the denominator by \( 4^n \).
We divide the numerator and denominator by \( 3^n \).
We substitute \(n=\infty \).
We take \( 3^n \) outside the brackets in the numerator and the denominator.
We take \( 4 \) outside the brackets in the numerator and the denominator.
1003047409
Level:
B
The sequence \( \left(\frac{2\cdot3^n+4^n+5}{4\cdot3^n-1}\right)_{n=1}^{\infty} \) is:
divergent and \( \lim\limits_{n\rightarrow\infty}\frac{2\cdot3^n+4^n+5}{4\cdot3^n-1}=\infty \)
convergent and \( \lim\limits_{n\rightarrow\infty}\frac{2\cdot3^n+4^n+5}{4\cdot3^n-1}=\frac12 \)
convergent and \( \lim\limits_{n\rightarrow\infty}\frac{2\cdot3^n+4^n+5}{4\cdot3^n-1}=\frac14 \)
convergent and \( \lim\limits_{n\rightarrow\infty}\frac{2\cdot3^n+4^n+5}{4\cdot3^n-1}=0 \)
divergent and it does not have an infinite limit
1003047410
Level:
B
Find the limit \(\lim\limits_{n\rightarrow\infty}\frac{2\cdot3^n+4^n+5}{3\cdot5^n-1} \).
\( 0 \)
\( \infty \)
\( \frac23 \)
\( -5 \)
\( 4 \)
1003047503
Level:
B
Choose the sequence with the limit equal to \( -5 \).
\( (0.15^n-5)_{n=1}^{\infty} \)
\( ((-5)^n-0.15)_{n=1}^{\infty} \)
\( (5^n-5)_{n=1}^{\infty} \)
\( (5^n+5)_{n=1}^{\infty} \)
\( (-0.15^n+5)_{n=1}^{\infty} \)
1003047507
Level:
B
Find the following limit. \[ \lim\limits_{n\rightarrow\infty}\frac{4^{2n}}{4^n+2} \]
\( \infty \)
\( 0 \)
\( 1 \)
\( 4 \)
\( 16 \)
1003047508
Level:
B
Find the following limit. \[ \lim\limits_{n\rightarrow\infty}\frac{\left(\frac13\right)^n}{\left(\frac15\right)^n-\left(\frac14\right)^n} \]
\( -\infty \)
\( \infty \)
\( 0 \)
\( 1 \)
\( -1 \)