1003109909
Level:
B
Evaluate the limit.
\[ \lim\limits_{x\to\infty}\!\left(\sqrt{x-3}-\sqrt x\right) \]
\( 0 \)
\( \infty \)
\( -\infty \)
\( -3 \)
Limits and continuity
1003109910
Level:
B
Evaluate the limit.
\[ \lim\limits_{x\to-\infty}\!\left(2+x+\sqrt{x^2-1}\right) \]
\( 2 \)
\( \infty \)
\( -\infty \)
\( 0 \)
1003109911
Level:
B
Evaluate the limit.
\[ \lim\limits_{x\to\infty}\!\left(\sqrt{2x+9x^2}-3x\right) \]
\( \frac13 \)
\( \frac23 \)
\( 0 \)
\( \infty \)
1003109912
Level:
B
Evaluate the limit.
\[ \lim\limits_{x\to\infty}\!\left( x-\sqrt{x^2-x+1} \right) \]
\( \frac12 \)
\( 1 \)
\( 0 \)
\( -\infty \)
\( -\frac12 \)
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 \(x = 3\).
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.
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.
2010001801
Level:
B
Find all points of discontinuity of the function \( f(x)=\frac{x^2-2x-8}{x^2-7x+12} \).
\( x_1 = 3\), \( x_2 = 4 \)
\( x_1 = -3 \), \( x_2 = -4 \)
\( x_1 = 3 \)
There is no point of discontinuity.
2010001802
Level:
B
Find all points of discontinuity of the function \( f(x)=\frac{9-x^2}{x^2+2x+3} \).
There is no point of discontinuity.
\( x_1 = 1\), \( x_2 =3 \)
\( x_1 = -1\), \( x_2 = -3 \)
\( x_1 = 1\)
2010001803
Level:
B
Evaluate the limit.
\[ \lim\limits_{x\to\infty}\frac{\sqrt x-2x}{x+3} \]
\(-2\)
\(3\)
\(2\)
\(-1\)