1003047403
Level:
B
Find the limit \( \lim\limits_{n\rightarrow\infty}\frac{5^{n+1}+6^{n+1}}{5^{n+1}+6^n} \).
\( 6 \)
\( 5 \)
\( \frac56 \)
\( 0 \)
\( \infty \)
B
1003047402
Level:
B
Choose the best first step to simplify and calculate the limit of the following sequence. \[ \left(\frac{3\cdot5^n+2\cdot6^n}{2\cdot5^n+4\cdot6^n}\right)_{n=1}^{\infty} \]
We take \( 6^n \) outside the brackets in the numerator and the denominator.
We take \( 5^n \) outside the brackets in the numerator and denominator.
We divide the numerator and denominator by \( 5^n \).
We divide the numerator by \( 6^n \).
We divide the denominator by \( 6^n \).
1003047401
Level:
B
Choose the correct formula to calculate the limit. \[ L=\lim\limits_{n\rightarrow\infty}\frac{3\cdot5^n+2\cdot6^n}{2\cdot5^n+4\cdot7^n } \]
\( L=\lim\limits_{n\rightarrow\infty} \frac{3\cdot\left(\frac57\right)^n+2\cdot\left(\frac67\right)^n}{2\cdot\left(\frac57\right)^n+4} =0 \)
\( L=\lim\limits_{n\rightarrow\infty}\frac{3\cdot\left(\frac56\right)^n+2}{2\cdot\left(\frac57\right)^n+4}=\frac12 \)
\( L=\lim\limits_{n\rightarrow\infty}\frac{3+2\cdot\left(\frac65\right)^n}{2+4\cdot\left(\frac75\right)^n } =\frac32 \)
\( L=\frac{3\cdot5^{\infty}+2\cdot6^{\infty}}{2\cdot5^{\infty}+4\cdot7^{\infty}}=\infty \)
\( L=\lim\limits_{n\rightarrow\infty}\frac{3\cdot\left(\frac57\right)^n+2\cdot\left(\frac67\right)^n}{2\cdot\left(\frac57\right)^n+4\cdot\left(\frac77\right)^n}=\frac56 \)
1103118505
Level:
B
Suppose a logarithmic function \( f \) is unbounded, increasing and \( f \) has an asymptote at \( x=-2 \). Given properties of \( f \), identify a possible graph of this function.
1103118504
Level:
B
Given the next graph of a logarithmic function \( f \), choose the correct list of properties of \( f \).
decreasing, asymptote at \( x=0 \), unbounded
decreasing, asymptote at \( y=0 \), unbounded
decreasing, asymptote at \( x=0 \), bounded below
increasing, asymptote at \( y=0 \), bounded below
1003118503
Level:
B
Which of the following functions is increasing?
\( f\colon y=-\log_{\frac12}x \)
\( f\colon y=-\log_2x \)
\( f\colon y=-2\log_2x \)
\( f\colon y=\log_{\frac12}x \)
1003118502
Level:
B
Given a function \( f(x)=b\cdot \log_ax \), where \( a > 1 \) and \( b < 0 \), find the correct statement.
Function \( f \) is decreasing.
Function \( f \) is increasing.
Function \( f \) is bounded.
Function \( f \) is bounded below.
1003118501
Level:
B
Given a decreasing logarithmic function \( f(x)=\log_ax \), choose the correct statement about the base \( a \).
\( 0 < a < 1 \)
\( a > 1 \)
\( a=1 \)
\( a < 1 \)
1003023309
Level:
B
What angle sweeps the hour hand of a clock in \( 3 \) hours?
\( -90^{\circ} \)
\( 45^{\circ} \)
\( -45^{\circ} \)
\( 90^{\circ} \)
1003023308
Level:
B
What angle sweeps the minute hand of a clock in \( 10 \) minutes?
\( -60^{\circ} \)
\( 36^{\circ} \)
\( -36^{\circ} \)
\( 60^{\circ} \)