Question:

Consider the function

1. Determine the limit of $f(x)$ as $x$ approaches positive infinity.
2. Determine the limit of $f(x)$ as $x$ approaches negative infinity.
3. Determine whether the function $f(x)$ has a horizontal asymptote. If so, find its equation.

Show all your work and justify your answers.

Answer:

1. To find the limit of $f(x)$ as $x$ approaches positive infinity, we consider the behavior of the function as $x$ becomes large. We divide the numerator and denominator of $f(x)$ by the highest power of $x$ to simplify the expression:

As $x$ approaches infinity, both $\frac{5}{x}$ and $\frac{2}{x}$ approach zero. Therefore, the expression simplifies to:

Thus, the limit of $f(x)$ as $x$ approaches positive infinity is $\boxed{\infty}$.

2. To find the limit of $f(x)$ as $x$ approaches negative infinity, we once again divide the numerator and denominator of $f(x)$ by the highest power of $x$:

As $x$ approaches negative infinity, both $\frac{5}{x}$ and $\frac{2}{x}$ approach zero. So, the expression simplifies to:

The limit of $f(x)$ as $x$ approaches negative infinity is $\boxed{-\infty}$.

3. To determine whether $f(x)$ has a horizontal asymptote, we examine the limit of $f(x)$ as $x$ approaches positive infinity and as $x$ approaches negative infinity.

From the previous parts, we found that as $x$ approaches positive infinity, $f(x)$ approaches infinity, and as $x$ approaches negative infinity, $f(x)$ approaches negative infinity. Since these limits are different, $f(x)$ does not have a horizontal asymptote.

Hence, the function $f(x)$ does not have a horizontal asymptote.