1003021113
Level:
B
Evaluate the following limit.
\[ \lim_{x\to0}\frac{\sin \frac x2}x \]
\( \frac12 \)
\( 2 \)
\( 1 \)
\( 0 \)
Limits and continuity
1003085501
Level:
B
Decide which of the following functions are continuous at \( x = 1 \).
\[\begin{aligned} f_1(x)&=\frac{x^2+1}{x-1} \\
f_2(x)&=\sqrt{x-1} \\
f_3(x)&=\log x \\
f_4(x)&=\mathrm{tg}(x-1)
\end{aligned}\]
The only such functions are:
\( f_3 \), \( f_4 \)
\( f_2 \), \( f_3 \), \( f_4 \)
\( f_2 \), \( f_3 \)
\( f_3 \)
1003085502
Level:
B
Determine the value of \( a \) (\(a\in\mathbb{R}\)) for which the function \( f(x)=\frac{ax}{x^2+a} \) is not continuous at \( x = -1 \).
\( -1 \)
\( 1 \)
\( 0 \)
\( 1\), \( -1 \)
1003085503
Level:
B
Determine the value of \( a \) (\(a\in\mathbb{R}\)) for which the function \( f(x)=\frac{x+a}{ax^2-3} \) is not continuous at \( x = 1 \).
\( 3 \)
\( -3 \)
\( \sqrt3 \)
\( -\sqrt3 \)
1003085504
Level:
B
Determine the value of \( a \) (\(a\in\mathbb{R}\)) for which the function
\[ f(x)=\left\{ \begin{array}{lc} x^2+x & \text{if } x\leq -2 \\ ax+3 & \text{if }x > -2 \end{array}\right. \]
is continuous at \( x = -2 \).
\( \frac12 \)
\( -\frac12 \)
\( \frac32 \)
\( -\frac32 \)
1003085505
Level:
B
Decide which of the following functions have points of discontinuity.
\[ \begin{aligned}
f_1(x)&=\frac{x-1}{x^2+1} \\
f_2(x)&=\frac{x^2-1}{x+1} \\
f_3(x)&=\frac{3x-1}{x^3+1} \\
f_4(x)&=\frac{x+1}{x^2-x+1}
\end{aligned} \]
The only such functions are:
\( f_2 \), \( f_3 \)
\( f_1 \), \( f_2 \), \( f_3 \)
\( f_2 \), \( f_3 \), \( f_4 \)
\( f_2 \), \( f_4 \)
1003085506
Level:
B
Decide which of the following functions have points of discontinuity.
\[ \begin{aligned}
f_1(x)&=\frac1{2^x} \\
f_2(x)& =x\cdot3^x \\
f_3(x)&=\frac1{e^x-1} \\
f_4(x)& =e^{2x-1}
\end{aligned} \]
The only such functions are:
\( f_3 \)
\( f_1 \), \( f_3 \)
\( f_1 \), \( f_4 \)
\( f_1 \), \( f_3 \), \( f_4 \)
1003085507
Level:
B
Decide which of the following functions do not have points of discontinuity.
\[ \begin{aligned}
f_1(x)&=\left\{\begin{array}{lc} x^2 & \text{if } x\leq 1 \\ 2x & \text{ if } x > 1 \end{array} \right. \\
f_2(x)& =\left\{ \begin{array}{lc} x^2-2x & \text{if } x < -1 \\ 3x & \text{if } x\geq-1 \end{array} \right. \\
f_3(x)&=\left\{ \begin{array}{lc} 3-x & \text{if } x\leq 2 \\ (x-1)^2 & \text{if } x > 2 \end{array} \right. \\
f_4(x)&=\left\{ \begin{array}{lc}x^2-2x+1& \text{if } x < 1 \\ \sqrt{x-1} & \text{if } x\geq1 \end{array} \right.
\end{aligned} \]
The only such functions are:
\( f_3 \), \( f_4 \)
\( f_2 \), \( f_3 \), \( f_4 \)
\( f_2 \), \( f_3 \)
\( f_3 \)
1003085508
Level:
B
Find all points of discontinuity of the function \( f(x)=\frac{x-3}{x^2-2x-3} \).
\( x_1=-1\), \( x_2 = 3 \)
\( x_1=-1\), \( x_2 = -3 \)
\( x_1=-1\)
There is no point of discontinuity.
1003085509
Level:
B
Find all points of discontinuity of the function \( f(x)=\frac{x^2+4x-5}{x^2+7x+10} \).
\( x_1 = -2 \), \( x_2 = -5 \)
\( x_1 = 2 \), \( x_2 = 5 \)
\( x_1 = -2 \)
There is no point of discontinuity.