9000141909
Level:
A
Given the function \(h\),
find \(\lim _{x\to \infty }h(x)\).
\[
h(x)=\begin{cases}
-\frac1{x-1} & \text{if } x< 1,\\
-(x-1)^2+2 & \text{if } x\geq 1
\end{cases}
\]
\(-\infty \)
\(1\)
\(0\)
\(\infty \)
Does not exist
Limits and Continuity of Functions
9000141910
Level:
A
Given the function \(h\),
find \(\lim _{x\to -\infty }h(x)\).
\[
h(x)=\begin{cases}
-\frac1{x-1} & \text{if } x< 1,\\
-(x-1)^2+2 & \text{if } x\geq 1
\end{cases}
\]
\(0\)
\(2\)
\(\infty \)
\(-\infty \)
Does not exist
1003021101
Level:
B
Evaluate the following limit.
\[ \lim_{x\to -2}\frac{2-\sqrt{x+6}}{x-2} \]
\( 0 \)
\( \frac14 \)
\( -\frac14 \)
\( 1 \)
1003021102
Level:
B
Evaluate the following limit.
\[ \lim_{x\to\frac{\pi}4}\frac{1+\sin2x}{1-\cos4x} \]
\( 1 \)
\( 0 \)
\( 2 \)
\( \frac12 \)
1003021107
Level:
B
Evaluate the following limit.
\[ \lim_{x\to\pi}\frac{\mathrm{tg}\,x}{\sin2x} \]
\( \frac12 \)
\( -\frac12 \)
\( 0 \)
\( -2 \)
1003021108
Level:
B
Evaluate the following limit.
\[ \lim_{x\to\frac{\pi}4}\frac{\cos x-\sin x}{1-\mathrm{cotg}\,x} \]
\( -\frac{\sqrt2}{2} \)
\( \frac{\sqrt2}{2} \)
\( 0 \)
\( -\sqrt2 \)
1003021109
Level:
B
Evaluate the following limit.
\[ \lim\limits_{x\rightarrow0}\frac{\sin2x}{x \cos x} \]
\( 2 \)
\( 1 \)
\( 0 \)
\( \frac12 \)
1003021110
Level:
B
Evaluate the following limit.
\[ \lim\limits_{x\rightarrow6}\frac{x-6}{\sqrt{x+3}-3} \]
\( 6 \)
\( 0 \)
\( 9 \)
\( 12 \)
1003021111
Level:
B
Evaluate the following limit.
\[ \lim_{x\to 3}\frac{9-x^2}{2-\sqrt{7-x}} \]
\( -24 \)
\( 0 \)
\( 24 \)
\( 36 \)
1003021112
Level:
B
Evaluate the following limit.
\[ \lim\limits_{x\rightarrow0}\frac{\sqrt{1+x^2}-1}x \]
\( 0 \)
\( 1 \)
\( \frac12 \)
\( 2 \)