1003163909
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to0}\frac{x-\sin x}{x^3} \]
\( \frac16 \)
\( -\frac16 \)
\( \frac13 \)
\( -\frac13 \)
\( 0 \)
Applications of derivatives
1003163908
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to1}\frac{\cos(\pi x)+1}{(x-1)^2} \]
\( \frac{\pi^2}2 \)
\( -\frac{\pi^2}2 \)
\( \frac{\pi}2 \)
\( -\frac{\pi}2 \)
1003163907
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to\infty}\frac{\mathrm{e}^x-2}{x^2} \]
\( \infty \)
\( 0 \)
\( 1 \)
\( \frac12 \)
1003163906
Level:
A
Evaluate the following limit. (Apply L'Hospital's rule repeatedly.)
\[ \lim_{x\to\infty}\frac{x^2-1}{x^3-2x^2+x} \]
\( 0 \)
\( \infty \)
\( \frac23 \)
\( 1 \)
1003163905
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to1}\frac{\sqrt{x+3}-2}{\ln x} \]
\( \frac14 \)
\( \frac12 \)
\( 0 \)
\( 1 \)
\( \frac{\sqrt2}2 \)
1003163904
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to\frac{\pi}3}\frac{1-2\cos x}{\pi-3x} \]
\( -\frac{\sqrt3}3 \)
\( -\frac{\sqrt3}6 \)
\( \frac{\sqrt3}3 \)
\( \frac{\sqrt3}6 \)
\( -\frac13 \)
1003163903
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to0}\frac{\sin2x}{\mathrm{tg}\,3x} \]
\( \frac23 \)
\( 1 \)
\( 2 \)
\( 0 \)
\( 6 \)
1003163902
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to0}\frac{\mathrm{e}^x-1}{\sin2x} \]
\( \frac12 \)
\( -\frac12 \)
\( 1 \)
\( 0 \)
\( -1 \)
1003163901
Level:
A
Use L'Hospital's rule to find the following limit.
\[ \lim_{x\to2}\frac{2x^3-3x^2-4}{x^2+x-6} \]
\( \frac{12}5 \)
\( \frac{18}5 \)
\( \frac{12}3 \)
\( 0 \)
\( \infty \)
9000145402
Level:
C
Identify a true statement about the function
\(f(x) = 2x^{2} -\frac{x^{4}}
{4} \).
The global maximum of \(f\)
on \(\mathbb{R}\) is
at \(x = 2\)
and \(x = -2\).
The global minimum of \(f\)
on \(\mathbb{R}\) is
at \(x = 2\)
and \(x = -2\).
The function \(f\) has a local
minimum at the point \(x = 2\).
The function \(f\) has a local
minimum at the point \(x = -2\).