To determine the end behavior of the given rational function
[tex]\[ f(x) = \frac{6x^5 + 8x^4 + 3x}{6x^2 - 4x - 5} \][/tex]
as [tex]\( x \)[/tex] approaches positive infinity ([tex]\( \infty \)[/tex]) and negative infinity ([tex]\( -\infty \)[/tex]), we need to look at the dominant terms in the numerator and the denominator.
### Step-by-Step Solution
1. Identify the highest-degree terms in the numerator and the denominator:
- The numerator is [tex]\( 6x^5 + 8x^4 + 3x \)[/tex]. The highest-degree term here is [tex]\( 6x^5 \)[/tex].
- The denominator is [tex]\( 6x^2 - 4x - 5 \)[/tex]. The highest-degree term here is [tex]\( 6x^2 \)[/tex].
2. Simplify the rational function by focusing on the highest-degree terms:
- For large [tex]\( x \)[/tex], the lower-degree terms become negligible in comparison to the highest-degree terms. Therefore, the function simplifies to:
[tex]\[
f(x) \approx \frac{6x^5}{6x^2}
\][/tex]
3. Simplify the expression:
- Simplifying the above fraction, we get:
[tex]\[
f(x) \approx \frac{6x^5}{6x^2} = x^3
\][/tex]
4. Determine the end behavior:
- As [tex]\( x \to \infty \)[/tex] (positive infinity), [tex]\( x^3 \to \infty \)[/tex]. Thus,
[tex]\[
\lim_{x \to \infty} f(x) = \infty
\][/tex]
- As [tex]\( x \to -\infty \)[/tex] (negative infinity), [tex]\( x^3 \to -\infty \)[/tex]. Thus,
[tex]\[
\lim_{x \to -\infty} f(x) = -\infty
\][/tex]
### Conclusion
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( f(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( f(x) \)[/tex] approaches negative infinity.
Therefore, the end behavior of [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) \rightarrow \infty \text{ as } x \rightarrow \infty \][/tex]
and
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty \][/tex]
