2010001804
Level:
B
Evaluate the following limit.
\[ \lim_{x\to\frac{\pi}4}\frac{\sin^2 x-\cos^2 x}{\mathrm{tg}\,x-1} \]
\(1\)
\( \frac{\sqrt2}{2} \)
\(0\)
\(-1\)
Limits and continuity
1003164301
Level:
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to3} f(x)=\infty\ \wedge\ \lim\limits_{x\to3} g(x)=\infty\ \wedge\ \lim\limits_{x\to3}\frac{f(x)}{g(x)}=5 \)
\( \lim\limits_{x\to3} f(x)=1\ \wedge\ \lim\limits_{x\to3} g(x)=\infty\ \wedge\ \lim\limits_{x\to3}\frac{f(x)}{g(x)}=5 \)
\( \lim\limits_{x\to3} f(x)=\infty\ \wedge\ \lim\limits_{x\to3} g(x)=1\ \wedge\ \lim\limits_{x\to3}\frac{f(x)}{g(x)}=5 \)
\( \lim\limits_{x\to3} f(x)=0\ \wedge\ \lim\limits_{x\to3} g(x)=\infty\ \wedge\ \lim\limits_{x\to3}\frac{f(x)}{g(x)}=5 \)
1003164302
Level:
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to2}f(x)=\infty\ \wedge\ \lim\limits_{x\to2}g(x)=\infty\ \wedge\ \lim\limits_{x\to2}[f(x)-g(x)]=\infty \)
\( \lim\limits_{x\to2}f(x)=1\ \wedge\ \lim\limits_{x\to2}g(x)=\infty\ \wedge\ \lim\limits_{x\to2}\frac{f(x)}{g(x)}=1 \)
\( \lim\limits_{x\to2}f(x)=-\infty\ \wedge\ \lim\limits_{x\to2}g(x)=1\ \wedge\ \lim\limits_{x\to2}[f(x)+g(x)]=1 \)
\( \lim\limits_{x\to2}f(x)=-\infty\ \wedge\ \lim\limits_{x\to2}g(x)=-\infty\ \wedge\ \lim\limits_{x\to2}[f(x)\cdot g(x)]=-\infty \)
1003164303
Level:
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to5}f(x)=0\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
\( \lim\limits_{x\to5}f(x)=1\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
\( \lim\limits_{x\to5}f(x)=\infty\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
\( \lim\limits_{x\to5}f(x)=-\infty\ \wedge\ \lim\limits_{x\to5}g(x)=\infty\ \wedge\ \lim\limits_{x\to5}[f(x)\cdot g(x)]=13 \)
1003164304
Level:
C
Which of the following situations could arise for suitable functions \( f \) and \( g \)?
\( \lim\limits_{x\to2} f(x)=\infty\ \wedge\ \lim\limits_{x\to2}g(x)=-\infty\ \wedge\ \lim\limits_{x\to2}[f(x)+g(x)]=-\infty \)
\( \lim\limits_{x\to2} f(x)=13\ \wedge\ \lim\limits_{x\to2}g(x)=0\ \wedge\ \lim\limits_{x\to2}\frac{f(x)}{g(x)}=13 \)
\( \lim\limits_{x\to2} f(x)=-\infty\ \wedge\ \lim\limits_{x\to2}g(x)=\infty\ \wedge\ \lim\limits_{x\to2}[f(x)-g(x)]=0 \)
\( \lim\limits_{x\to2} f(x)=\infty\ \wedge\ \lim\limits_{x\to2}g(x)=-\infty\ \wedge\ \lim\limits_{x\to2}[f(x)\cdot g(x)]=\infty \)