Graph the tunction. Give the equations of the vertical and horizontal asymptotes. $$ y=\frac{2 x}{\mathrm{x}-4} $$ Give the equations of any vertical asymptotes for the graph of the rational function. Select the correct choice and, if necessary, fil in the answer box(es) within your choice. A. The function has one vertical asymptote. The vertical asymptote is (Simplify your answer. Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is and the rightmost asymplote is (Simplify your answers. Type equations.) C. There is no vertical asymptote.

Question

UpStudy AI · Accepted Answer

**A. The function has one vertical asymptote. The vertical asymptote is**
$$ x = 4 $$

---

**Explanation:**

To determine the vertical asymptotes of the rational function $$ y = \frac{2x}{x - 4} $$, follow these steps:

1. **Identify where the denominator is zero:**
   $$
   x - 4 = 0 \implies x = 4
   $$
   The function is undefined at $$ x = 4 $$, which indicates a potential vertical asymptote.

2. **Check the numerator at $$ x = 4 $$:**
   $$
   2(4) = 8 \neq 0
   $$
   Since the numerator is not zero when $$ x = 4 $$, there is indeed a vertical asymptote at this value.

3. **Conclusion:**
   There is **one** vertical asymptote at $$ x = 4 $$.

---

**Additional Information:**

While the primary focus here is on vertical asymptotes, it's also useful to identify the horizontal asymptote for a complete understanding of the function's behavior.

- **Horizontal Asymptote:**
  $$
  \lim_{x \to \pm\infty} \frac{2x}{x - 4} = \lim_{x \to \pm\infty} \frac{2}{1 - \frac{4}{x}} = 2
  $$
  So, the horizontal asymptote is $$ y = 2 $$.

**Graph Overview:**
- **Vertical Asymptote:** $$ x = 4 $$
- **Horizontal Asymptote:** $$ y = 2 $$
  
The graph will approach these asymptotes but never actually touch them.
**A. The function has one vertical asymptote. The vertical asymptote is** $$ x = 4 $$

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Pre Calculus Questions