Limits and continuity

1003093101

Level: 
A
Which of the statements A, B, C, D, E given bellow are correct? \[ \begin{aligned} \text{A: }& \lim\limits_{x\to1^+}⁡\frac{2x}{x-1}=\infty \\ \text{B: }& \lim\limits_{x\to-2^+}⁡\frac{x+1}{x+2}=-\infty \\ \text{C: }& \lim\limits_{x\to3^-}\frac{x+1}{3-x}=-\infty \\ \text{D: }& \lim\limits_{x\to-1^+}\frac{x}{x^2-1}=\infty \\ \text{E: }& \lim\limits_{x\to2^-}\frac{x+3}{4-x^2} =\infty \end{aligned} \] The only correct statements are:
A, B, D, E
A, B, D
A, D, E
A, B, C, E
A, B, C
B, D, E

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:
\( f_3 \), \( f_4 \)
\( f_2 \), \( f_3 \), \( f_4 \)
\( f_2 \), \( f_3 \)
\( f_3 \)

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:
\( f_3 \)
\( f_1 \), \( f_3 \)
\( f_1 \), \( f_4 \)
\( f_1 \), \( f_3 \), \( f_4 \)

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:
\( f_2 \), \( f_3 \)
\( f_1 \), \( f_2 \), \( f_3 \)
\( f_2 \), \( f_3 \), \( f_4 \)
\( f_2 \), \( f_4 \)

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 \( x = -2 \).
\( \frac12 \)
\( -\frac12 \)
\( \frac32 \)
\( -\frac32 \)