Mathematical Definition of Horizontal Asymptotes

A horizontal asymptote of a function $f(x)$ is a horizontal line $y = c$ that the function approaches as $x \to \infty$ or $x \to -\infty$. Formally, if:

• $\lim_{x \to \infty} f(x) = c$, then $y = c$ is a horizontal asymptote as $x \to \infty$.
• $\lim_{x \to -\infty} f(x) = c$, then $y = c$ is a horizontal asymptote as $x \to -\infty$.

A function may have:

1. One horizontal asymptote at $y = c$ as $x \to \infty$ and another as $x \to -\infty$,
2. The same asymptote for both directions, or
3. No horizontal asymptotes at all if the limits do not exist or diverge.

Determining Horizontal Asymptotes for $f(x) = \frac{x^2 + 5x + 7}{4 + 5x^2}$

Let's analyze the function $f(x) = \frac{x^2 + 5x + 7}{4 + 5x^2}$ to determine whether it admits horizontal asymptotes.

Step 1: Compare the degrees of the polynomial in the numerator and denominator.

• The numerator is $x^2 + 5x + 7$, which has a degree of 2.
• The denominator is $4 + 5x^2$, which also has a degree of 2.

Step 2: Horizontal Asymptote Rule

For rational functions of the form $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials:

• If the degree of $P(x)$ is less than the degree of $Q(x)$, there is a horizontal asymptote at $y = 0$.
• If the degree of $P(x)$ is greater than the degree of $Q(x)$, there is no horizontal asymptote.
• If the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients of $P(x)$ and $Q(x)$.

Step 3: Apply to $f(x) = \frac{x^2 + 5x + 7}{4 + 5x^2}$

• The degree of the numerator ( $x^2 + 5x + 7$ ) is 2.
• The degree of the denominator ( $4 + 5x^2$ ) is 2.

Since the degrees of the numerator and denominator are equal, we compute the horizontal asymptote by finding the ratio of the leading coefficients of $x^2$ in the numerator and denominator.

• The leading coefficient of the numerator is 1 (from $x^2$).
• The leading coefficient of the denominator is 5 (from $5x^2$).

Thus, the horizontal asymptote is:

$y = \frac{1}{5}$

Conclusion

The function $f(x) = \frac{x^2 + 5x + 7}{4 + 5x^2}$ does admit a horizontal asymptote at $y = \frac{1}{5}$.

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Would you like further clarification or details about horizontal asymptotes? Here are some related questions you might find helpful:

1. How can I determine vertical asymptotes for rational functions?
2. What happens to a function if the degree of the numerator is greater than the denominator?
3. Can a rational function have both horizontal and vertical asymptotes simultaneously?
4. How do I find the end behavior of a function using asymptotes?
5. What are slant (oblique) asymptotes, and how do they differ from horizontal asymptotes?

Tip: For rational functions, the horizontal asymptote is especially useful in determining the behavior of the function as $x$ grows larger in magnitude, whether positive or negative.