Limits and Continuity of Functions

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

1103080004

Level: 
A
The graph of the function \( f \) is given in the figure. Choose the incorrect statement.
\( \lim\limits_{x\rightarrow1^-} f(x) = -1 \)
\( \lim\limits_{x\rightarrow -1} f(x) \) does not exist
\( \lim\limits_{x\rightarrow1^+} f(x) = 0 \)
\( \lim\limits_{x\rightarrow-\infty} f(x) = 1 \)

1103080003

Level: 
A
The graph of the function \( f \) is given in the figure. Choose the incorrect statement.
\( \lim\limits_{x\rightarrow \infty} f(x) = -x \)
\( \lim\limits_{x\rightarrow 0^+} f(x) = 0 \)
\( \lim\limits_{x\rightarrow 0^-} f(x) = \infty \)
\( \lim\limits_{x\rightarrow-\infty} f(x) = \infty \)

1103080002

Level: 
A
The graph of the function \( f \) is given in the figure. Choose the incorrect statement.
\( \lim\limits_{x\rightarrow-1}f(x) \) does not exist
\( \lim\limits_{x\rightarrow\infty} f(x) = \infty \)
\( \lim\limits_{x\rightarrow0} f(x) = 0 \)
\( \lim\limits_{x\rightarrow-\infty} f(x) = 1 \)

1103080001

Level: 
A
The graph of the function \( f \) is given in the figure. Choose the incorrect statement. Dashed lines represent asymptotes of the function $f$.
\( \lim\limits_{x\rightarrow \infty} f(x) = -\infty \)
\( \lim\limits_{x\rightarrow -2^-} f(x) = -\infty \)
\( \lim\limits_{x\rightarrow \infty} f(x) = -2 \)
\( \lim\limits_{x\rightarrow -2} f(x) \) does not exist