Limits and continuity

1003109904

Level: 
B
Choose the proper expression to expand \( \sqrt{x^2-2}-x \) when evaluating the limit \( \lim\limits_{x\to\infty}\!\left(\sqrt{x^2-2}-x+1\right) \).
\( \frac{\sqrt{x^2-2}+x}{\sqrt{x^2-2}+x} \)
\( \frac{\sqrt{x^2-2}-x}{\sqrt{x^2-2}-x} \)
\( \frac{\sqrt{x^2-2}}{\sqrt{x^2-2}} \)
\( \frac{\sqrt{x^2+2}+x}{\sqrt{x^2+2}+x} \)

1003109903

Level: 
B
Find the next possible correct step in evaluating this limit. \[ \lim\limits_{x\to\infty}\frac{2x^2+3}{\sqrt{3x^4-1}} \]
\( \lim\limits_{x\to\infty}\frac{2+\frac3{x^2}}{\sqrt{3-\frac1{x^4}}} \)
\( \lim\limits_{x\to\infty}\frac{2+\frac3{x^2}}{\sqrt{3x^2-\frac1{x^2}}} \)
\( \lim\limits_{x\to\infty}\frac{2+\frac3{x^2}}{\sqrt{3x^3-\frac1x}} \)
\( \lim\limits_{x\to\infty}\frac{\frac2{x^2}+\frac3{x^4}}{\sqrt{3-\frac1{x^4}}} \)

1003109902

Level: 
B
Find the next possible correct step in evaluating this limit. \[ \lim\limits_{x\to\infty}\frac{\sqrt{x^2+1}-x}{x+1} \]
\( \lim\limits_{x\to\infty}\frac{\sqrt{1+\frac1{x^2}}-1}{1+\frac1x} \)
\( \lim\limits_{x\to\infty}\frac{\sqrt{1+\frac1{x^2}}-\frac1x}{1+\frac1x} \)
\( \lim\limits_{x\to\infty}\frac{\sqrt{x+\frac1x}-1}{1+\frac1x} \)
\( \lim\limits_{x\to\infty}\frac{\sqrt{1+\frac1{x^2}}-\frac1x}{\frac1x+\frac1{x^2} } \)

1003109901

Level: 
B
Find the next possible correct step in evaluating this limit. \[ \lim\limits_{x\to\infty}\frac{2x-1}{\sqrt{2x^2-1}} \]
\( \lim\limits_{x\to\infty}⁡\frac{2-\frac1x}{\sqrt{2-\frac1{x^2}}} \)
\( \lim\limits_{x\to\infty}\frac{2-\frac1x}{\sqrt{2x-\frac1x}} \)
\( \lim\limits_{x\to\infty}\frac{2-\frac1x}{\sqrt{2-x}} \)
\( \lim\limits_{x\to\infty}\frac{\frac2x-\frac1{x^2}}{\sqrt{2-\frac1{x^2}}} \)

1003093102

Level: 
B
Which of the statements A, B, C, D, E given bellow are incorrect? \[ \begin{aligned} \text{A: } & \lim\limits_{x\to-\infty}\left(3-\frac1x\right)=3 \\ \text{B: } & \lim\limits_{x\to-\infty}\left(x^5-2\right)=\infty \\ \text{C: } & \lim\limits_{x\to-\infty}\left(0.3\cdot2^x\right)=-\infty \\ \text{D: } & \lim\limits_{x\to\infty}\left(0.5^x+5\right)= 5 \\ \text{E: } & \lim\limits_{x\to\infty}\left(\log_{\frac12}⁡x-x\right)=0 \end{aligned} \] The only incorrect statements are:
B, C, E
B, D
B, D, E
A, B, C
B, C