To find the horizontal asymptotes of the function, we need to analyze the behavior of the function as $x \to \infty$ and $x \to -\infty$.

Given function:

$f(x) = \frac{3(x+10)(x-10)}{2(x-10)}$

Step 1: Simplify the function

Notice that the factor $(x-10)$ appears in both the numerator and denominator, so we can cancel this factor (assuming $x \neq 10$):

$f(x) = \frac{3(x+10)}{2}$

Now the function simplifies to:

$f(x) = \frac{3(x+10)}{2} = \frac{3x + 30}{2}$

Step 2: Find the horizontal asymptote

To find horizontal asymptotes, we consider the limits of the function as $x \to \infty$ and $x \to -\infty$.

• As $x \to \infty$:

$f(x) = \frac{3x + 30}{2} \sim \frac{3x}{2} \quad \text{(since the constant term becomes negligible)}$

Thus, $f(x) \to \infty$ as $x \to \infty$. There is no horizontal asymptote at $x \to \infty$.

• As $x \to -\infty$:

$f(x) = \frac{3x + 30}{2} \sim \frac{3x}{2}$

Similarly, $f(x) \to -\infty$ as $x \to -\infty$, so there is no horizontal asymptote as $x \to -\infty$ either.

Conclusion:

The given function has no horizontal asymptotes, since in both cases as $x \to \infty$ and $x \to -\infty$, the function approaches infinity and negative infinity, respectively.

Would you like more details or have any questions?

Relative questions:

1. What are vertical asymptotes, and does this function have any?
2. Can we cancel terms in the function if they result in undefined values?
3. How do slant (oblique) asymptotes differ from horizontal ones?
4. What happens at $x = 10$ in the original function?
5. How can limits help in finding asymptotes for rational functions?

Tip:

When simplifying