Limits & Continuity — Quiz 2

Questions in this quiz (10)

1. $$\lim_{x\to\infty}\frac{\sqrt{x^6+3x^3+2}}{x^5+2x^4}$$

   • $$0$$
   • $$1$$
   • $$\infty$$
   • $$-1$$

2. Give the equations of the vertical and horizontal asymptotes of $$f(x)=\frac{3x^2-2x-8}{x^2-5x+6}$$

   • $$\text{VA: }x=2,3,\ \text{HA: }y=3$$
   • $$\text{VA: }x=2,3,\ \text{HA: }y=1$$
   • $$\text{VA: }x=2,\ \text{HA: }y=3$$
   • $$\text{VA: }x=3,\ \text{HA: }y=3$$

3. $$\lim_{x\to\infty}\frac{4x^3+2x^2+1}{\sqrt{5x^6+3x^2}}$$

   • $$\frac{4}{\sqrt{5}}$$
   • $$0$$
   • $$\infty$$
   • $$4$$

4. $$\lim_{x\to\infty}3\left(1+\frac{2}{x}\right)^x+\cos\left(\frac{2\pi}{x}\right)$$

   • $$3e^2+1$$
   • $$3e+1$$
   • $$3e^2$$
   • $$3e$$

5. $$\lim_{x\to0}\frac{2\cos x-2}{x^2}$$

   • $$-1$$
   • $$0$$
   • $$1$$
   • $$2$$

6. Give the equations of the horizontal and vertical asymptotes of $$g(x)=\frac{4a^2x^2+3abx-b^2}{8a^2x^2-2abx-b^2}$$

   • $$\text{HA: }y=\frac{1}{2}$$
   • $$\text{HA: }y=2$$
   • $$\text{HA: }y=4$$
   • $$\text{HA: }y=1$$

7. $$\lim_{x\to\infty}\left(\sqrt{9x^2+4x+1}-3x\right)$$

   • $$\frac{2}{3}$$
   • $$0$$
   • $$\infty$$
   • $$1$$

8. The population $$P$$ of bacteria is modeled by $$P(t)=40e^{0.15t}$$ Use the Intermediate Value Theorem to verify that the bacteria population reaches $$120$$ colonies on the interval $$[2,6]$$. Which statement justifies the Intermediate Value Theorem?

   • $$P(2)$$ < $$120$$ and P(6)$$ > $$120$$
   • $$P(2)$$ > $$120$$ and $$P(6)$$ > $$120$$
   • $$P(2)$$ < $$120$$ and $$P(6)$$ < $$120$$
   • $$P(2)=120$$

9. $$\lim_{x\to0}\sqrt{3+\arctan\left(\frac{1}{x}\right)}$$

   • $$\sqrt{3-\frac{\pi}{2}}$$
   • $$\sqrt{3+\frac{\pi}{2}}$$
   • $$\infty$$
   • does not exist

10. If $$y=\frac{1}{2+10^{1/x}}$$, find $$\lim_{x\to0}y$$

    • $$0$$
    • $$\frac{1}{12}$$
    • $$\frac{1}{2}$$
    • does not exist