Let's analyze the function $f(x) = \frac{7x^2 + 2x + 2}{2x - 1}$ and determine the asymptotes.

Step 1: Vertical Asymptotes

Vertical asymptotes occur where the denominator $2x - 1 = 0$, provided the numerator does not also equal zero at those points.

Set the denominator to zero: $2x - 1 = 0 \implies x = \frac{1}{2}.$

Thus, there is a vertical asymptote at: $x = \frac{1}{2}.$

Step 2: Oblique (Slant) Asymptote as $x \to \infty$ or $x \to -\infty$

For rational functions where the degree of the numerator is exactly one higher than the degree of the denominator, there is a slant (oblique) asymptote.

Perform polynomial long division for: $\frac{7x^2 + 2x + 2}{2x - 1}.$

Division process:

1. Divide the leading term: $\frac{7x^2}{2x} = \frac{7}{2}x$.
2. Multiply $\frac{7}{2}x$ by $2x - 1$: $\frac{7}{2}x \cdot (2x - 1) = 7x^2 - \frac{7}{2}x.$
3. Subtract: $(7x^2 + 2x + 2) - (7x^2 - \frac{7}{2}x) = \frac{11}{2}x + 2.$
4. Divide again: $\frac{\frac{11}{2}x}{2x} = \frac{11}{4}$.
5. Multiply $\frac{11}{4}$ by $2x - 1$: $\frac{11}{4}(2x - 1) = \frac{11}{2}x - \frac{11}{4}.$
6. Subtract: $\left( \frac{11}{2}x + 2 \right) - \left( \frac{11}{2}x - \frac{11}{4} \right) = \frac{19}{4}.$

The quotient is: $f(x) = \frac{7}{2}x + \frac{11}{4} + \frac{\frac{19}{4}}{2x - 1}.$

As $x \to \infty$, the remainder term $\frac{\frac{19}{4}}{2x - 1} \to 0$. Hence, the slant asymptote is: $y = \frac{7}{2}x + \frac{11}{4}.$

Final Answers:

1. Vertical Asymptote: $x = \frac{1}{2}$.
2. No second vertical asymptote.
3. Oblique Asymptote as $x \to \infty$: $y = \frac{7}{2}x + \frac{11}{4}$.
4. No horizontal asymptotes, as the degree of the numerator exceeds that of the denominator.

Would you like to see detailed steps for the long division? Let me know!

Related Questions:

1. What is the behavior of the function near $x = \frac{1}{2}$?
2. How would you find horizontal asymptotes for similar functions?
3. Can a function have both horizontal and slant asymptotes? Why or why not?
4. How does the numerator's degree affect the type of asymptote?
5. What changes in asymptotes occur if the numerator and denominator share common roots?

Tip:

When finding oblique asymptotes, always verify the degree difference between numerator and denominator.