Review - Fall $$ 2024-5 \% $$ Find the horizontal asymptote, if any, of the graph of the rational function. $$ h(x)=\frac{20 x^{3}}{5 x^{2}+1} $$

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To determine the **horizontal asymptote** of the rational function:

$$
h(x) = \frac{20x^{3}}{5x^{2} + 1}
$$

we analyze the degrees of the numerator and the denominator.

### Step 1: Identify the Degrees
- **Numerator:** $$20x^{3}$$ has a degree of **3**.
- **Denominator:** $$5x^{2} + 1$$ has a degree of **2**.

### Step 2: Compare the Degrees
- **Case 1:** If the degree of the **numerator** is **less than** the degree of the **denominator**, the horizontal asymptote is $$y = 0$$.
- **Case 2:** If the degrees are **equal**, the horizontal asymptote is the ratio of the leading coefficients.
- **Case 3:** If the degree of the **numerator** is **greater than** the degree of the **denominator**, there is **no horizontal asymptote**. Instead, the function may have an **oblique (slant) asymptote**.

### Applying to $$h(x)$$:
- The **degree of the numerator (3)** is **greater** than the **degree of the denominator (2)**.

### Conclusion:
Since the degree of the numerator is greater than the degree of the denominator, the function $$h(x)$$ **does not have a horizontal asymptote**.

**Answer:**  
There is no horizontal asymptote for the function $$ h(x) = \frac{20x^{3}}{5x^{2} + 1} $$.
The function $$ h(x) = \frac{20x^{3}}{5x^{2} + 1} $$ does not have a horizontal asymptote.

Review - Fall
Find the horizontal asymptote, if any, of the graph of the rational function.

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Review - Fall Find the horizontal asymptote, if any, of the graph of the rational function.