1003047605
Level:
C
Find the limit \( \lim\limits_{n\to\infty} \left( \sqrt n-\sqrt{n-1} \right) \).
\( 0 \)
\( \infty \)
\( -\infty \)
\( -1 \)
\( \sqrt2 \)
C
1003047604
Level:
C
Choose the correct computation of the limit. \[ L=\lim\limits_{n\to\infty} \left( \sqrt{n^2+3n}-2n \right) \]
\( L=\lim\limits_{n\to\infty}n\left( \sqrt{1+\frac3n}-2 \right) = -\infty \)
\( L= \infty-\infty=0 \)
\( L=\lim\limits_{n\to\infty}(n-2n)=-\infty \)
\( L=\lim\limits_{n\to\infty} \left( n^2+3n-4n^2 \right) =-3 \)
\( L=\lim\limits_{n\to\infty}\frac{n^2+3n-4n^2}{\sqrt{n^2+3n}+2n}=\infty \)
1003047603
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\left( \sqrt{4n^2+3n}-2n \right) \).
\( \frac34 \)
\( \infty \)
\( 0 \)
\( -\infty \)
\( \sqrt2 \)
1003047602
Level:
C
Choose the step to take first to efficiently evaluate the limit of the sequence \( \left(n-\sqrt{n^2-1} \right)_{n=1}^{\infty} \).
We expand with the expression \( n+\sqrt{n^2-1} \).
We expand with the expression \( n-\sqrt{n^2-1} \).
We expand with \( n \).
We multiply by the expression \( n+\sqrt{n^2-1} \).
We multiply by the expression \( n-\sqrt{n^2-1} \).
We substitute \( n=\infty \).
1003047601
Level:
C
Find the limit \( \lim\limits_{n\to\infty}\left(n-\sqrt{n-1}\right) \).
\( \infty \)
\( 0 \)
\( -\infty \)
\( 1 \)
\( \frac12 \)
1103107104
Level:
C
In one of the following pictures is the graph of the function \( f \), which is not one-to-one function. Choose the picture.
1103107103
Level:
C
The graph of the function \( f(x) \) and its inverse \( f^{-1} (x) \) is shown in one of the following pictures. Choose the picture.
1103107102
Level:
C
The graph of the function \( f(x) \) and its inverse \( f^{-1} (x) \) is shown in one of the following pictures. Choose the picture.
1003107101
Level:
C
Complete the following true statement: The graph of a function \( f(x) \) is the reflection the graph of its inverse \( f^{-1}(x) \)
across the line \( y=x \).
across the axis \( x \).
across the axis \( y \).
about the origin.
1003083110
Level:
C
The graphs of the quadratic functions \( f \) and \( g \) have not the same vertex and \( f(x)=ax^2+bx+c \), where \( a \), \( b \), \( c \) are nonzero real numbers. Find \( g(x) \) such that the graph of \( g \) is the reflection of the graph of \( f \) about \( y \)-axis.
\( g(x)=ax^2-bx+c \), i.e. the equation of \( f \) and \( g \) differ in the sign of the coefficient at the linear term only
\( g(x)=-ax^2+bx+c \), i.e the equation of \( f \) and \( g \) differ in the sign of the coefficient at the quadratic term only
\( g(x)=ax^2+bx-c \), i.e. the equation of \( f \) and \( g \) differ in the sign of the coefficient at the absolute term only
\( g(x)=-ax^2-bx-c \), i.e. \( g(x)=-f(x) \)
None of the statements above is true.