Limits and continuity
2000018703
Level:
B
The picture shows a graph of a function. Decide at which of the marked points \(x_1\), \(x_2\), \(x_3\) and \(x_4\), the left-hand and right-hand limit of the function has the same value. (Note: The dashed lines are asymptotes of the function.)
Only at \(x_1\) and \(x_3\).
Only at \(x_1\).
Only at \(x_3\).
The left-hand and right-hand limit is the same at any marked point.
2000018702
Level:
B
Choose the true statement about the limits of the function whose graph is shown in the picture. (Note: The dashed lines are asymptotes of the function.)
The function has the limit "negative infinity" only at \(x_2\) and at "negative infinity" it has the limit \(a_2\).
The function has the limit "negative infinity" at \(x_2\) and \(x_3\) and at "negative infinity" it has the limit \(a_2\).
The function has the limit "negative infinity" only at \(x_2\) and at "negative infinity" there is no limit.
The function has the limit "negative infinity" at \(x_2\) and \(x_3\) and at "negative infinity" there is no limit.
2000018701
Level:
B
The following pictures show graphs of \(3\) functions. Choose the true statement about the limit at \(x = 3\).
The functions \(f\), \(g\), \(h\) have the same limit at \(x = 3\).
The function \(g\) has no limit at \(x = 3\).
The function \(f\) has no limit at \(x = 3\).
The limits of functions \(f\), \(g\), \(h\) at \(x = 3\) differ.
Only the function \(h\) has a limit at \(x = 3\).
2010012704
Level:
A
Evaluate the following limit.
\[ \lim\limits_{x\to 1}\frac{1-x^3}{x^2+3x-4} \]
\( -\frac35\)
\( -3\)
\( \frac35\)
\(0\)
2010012708
Level:
A
Given the function \(g\) (see the picture),
find \(\lim\limits_{x\to -2^{+}}g(x)\).
\[
g(x)=\begin{cases}
\frac1{x+4}-1 & \text{if } x< -4,\\
\frac1{x+2}+1 & \text{if } x > -2
\end{cases}
\]
\( \infty\)
\(- \infty\)
\(1\)
\( -2\)
This limit does not exist.
2010012707
Level:
A
Given the function \(g\) (see the picture),
find \(\lim\limits_{x\to \infty}g(x)\).
\[
g(x)=\begin{cases}
\frac12(x-1)^2+1 & \text{if } x < 1,\\
\frac1{x^2}+2 & \text{if } x \geq 1
\end{cases}
\]
\(2\)
\( \infty\)
\( -\infty\)
\( 0\)
This limit does not exist.
2010012706
Level:
A
Given the function \(g\) (see the picture),
find \(\lim\limits_{x\to 1}g(x)\).
\[
g(x)=\begin{cases}
\frac12(x-1)^2+1 & \text{if } x < 1,\\
\frac1{x^2}+2 & \text{if } x \geq 1
\end{cases}
\]
This limit does not exist.
\( 3\)
\( 2\)
\( 1\)
2010012705
Level:
A
Given the graph of the function \( f \), find \( \lim\limits_{x\rightarrow 2}f(x) \).
\( 1\)
\( 4\)
\( 2\)
This limit does not exist.
2010012703
Level:
A
Evaluate the following limit.
\[ \lim\limits_{x\to 1}\frac{x^2+4x-5}{x^2-4x+3} \]
\( -3\)
\( -\frac53\)
\( 3\)
\(\frac53\)