2010012701
Level:
A
Evaluate the following limit.
\[ \lim\limits_{x\to-\infty}\left(-x^4+2x-5\right) \]
\( -\infty\)
\( \infty\)
\( -5\)
\(0\)
Limits and continuity
2010001804
Level:
B
Evaluate the following limit.
\[ \lim_{x\to\frac{\pi}4}\frac{\sin^2 x-\cos^2 x}{\mathrm{tg}\,x-1} \]
\(1\)
\( \frac{\sqrt2}{2} \)
\(0\)
\(-1\)
2010001803
Level:
B
Evaluate the limit.
\[ \lim\limits_{x\to\infty}\frac{\sqrt x-2x}{x+3} \]
\(-2\)
\(3\)
\(2\)
\(-1\)
2010001802
Level:
B
Find all points of discontinuity of the function \( f(x)=\frac{9-x^2}{x^2+2x+3} \).
There is no point of discontinuity.
\( x_1 = 1\), \( x_2 =3 \)
\( x_1 = -1\), \( x_2 = -3 \)
\( x_1 = 1\)
2010001801
Level:
B
Find all points of discontinuity of the function \( f(x)=\frac{x^2-2x-8}{x^2-7x+12} \).
\( x_1 = 3\), \( x_2 = 4 \)
\( x_1 = -3 \), \( x_2 = -4 \)
\( x_1 = 3 \)
There is no point of discontinuity.
Limits of Function from its Graph II
Submitted by ladislav.foltyn on Wed, 05/01/2019 - 12:19Question:
\vspace{-1em}
\begin{minipage}{0.3\linewidth}
The figure shows a~graph of a~function $f$. Mark the true values of the~given limits of $f$.
\end{minipage}
\hfill
\begin{minipage}{0.3\linewidth}
\obrA
\end{minipage}
\hfill
\begin{minipage}{0.35\linewidth}
\footnotesize
$$\def\arraystretch{2.2}
f(x)=\left\{\begin{array}{ll}-\frac1{x-\frac12}-1; & x < 0 \\ \mathrm{cotg}\!\left(\pi\frac x2\right); & 0\leq x < 2 \\ -\frac1{x-2}-6x+20; & 2\leq x < 3.2 \\ \sin \left(2\pi(x+0.3)\right) & x \geq 3.2 \end{array}\right.$$
\end{minipage}
Limits of Function from its Graph I
Submitted by ladislav.foltyn on Sat, 04/20/2019 - 12:00Question:
\vspace{-1em}
\begin{minipage}{0.3\linewidth}
The figure shows a~graph of a~function $f$. Mark the true values of the~given limits of $f$.
\end{minipage}
\hfill
\begin{minipage}{0.3\linewidth}
\centering
\obrA
\end{minipage}
\hfill
\begin{minipage}{0.35\linewidth}
\footnotesize
$$\def\arraystretch{2.2}
f(x)=\left\{\begin{array}{ll}-\frac1{x+1}+x+3; & x < -1 \\ \frac2{x+1}; & -1\leq x < 1 \\ -\frac2x+2; & x\geq 1 \end{array}\right.$$
\end{minipage}
1103080005
Level:
A
The graph of the function \( f \) is given in the figure bellow. Choose the incorrect statement.
\( \lim\limits_{x\to-\infty}f(x)=-1 \)
\( \lim\limits_{x\to\infty}f(x)=\infty \)
\( \lim\limits_{x\to1^-}f(x)=2 \)
\( \lim\limits_{x\to-1^+}f(x)=-\infty \)
1003164304
Level:
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to2} f(x)=\infty\ \wedge\ \lim\limits_{x\to2}g(x)=-\infty\ \wedge\ \lim\limits_{x\to2}[f(x)+g(x)]=-\infty \)
\( \lim\limits_{x\to2} f(x)=13\ \wedge\ \lim\limits_{x\to2}g(x)=0\ \wedge\ \lim\limits_{x\to2}\frac{f(x)}{g(x)}=13 \)
\( \lim\limits_{x\to2} f(x)=-\infty\ \wedge\ \lim\limits_{x\to2}g(x)=\infty\ \wedge\ \lim\limits_{x\to2}[f(x)-g(x)]=0 \)
\( \lim\limits_{x\to2} f(x)=\infty\ \wedge\ \lim\limits_{x\to2}g(x)=-\infty\ \wedge\ \lim\limits_{x\to2}[f(x)\cdot g(x)]=\infty \)
1003164303
Level:
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to5}f(x)=0\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
\( \lim\limits_{x\to5}f(x)=1\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
\( \lim\limits_{x\to5}f(x)=\infty\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
\( \lim\limits_{x\to5}f(x)=-\infty\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)