To describe the end behavior of the graph of the function [tex]\( f(x) = x^3(x+3)(-5x+1) \)[/tex] using limits, we need to analyze what happens to [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] tends to [tex]\(-\infty\)[/tex] and [tex]\( +\infty \)[/tex].
### Step-by-Step Analysis:
1. Identify the leading term:
We first note that the highest degree term in the polynomial will dominate the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\(\pm \infty\)[/tex].
2. Expand the polynomial:
We consider the most significant terms when multiplying out [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x^3 (x + 3) (-5x + 1) \][/tex]
Without full expansion, it's already clear that the leading term of this product will involve [tex]\( x^5 \)[/tex].
3. Determine the sign of the leading term:
The dominant term as [tex]\( x \)[/tex] becomes very large (positive or negative) will be the term with the highest power of [tex]\( x \)[/tex]. From [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) \approx -5x^5 \][/tex]
This means we should consider [tex]\( -5x^5 \)[/tex] for large [tex]\( x \)[/tex].
### Analyzing Limits for End Behavior:
4. As [tex]\( x \rightarrow +\infty \)[/tex]:
[tex]\[ \lim_{{x \to +\infty}} f(x) = \lim_{{x \to +\infty}} (-5x^5) \][/tex]
Since [tex]\( x^5 \)[/tex] grows very large and positive as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex]:
[tex]\[ -5x^5 \rightarrow -\infty \][/tex]
Therefore:
[tex]\[ \lim_{{x \to +\infty}} f(x) = -\infty \][/tex]
5. As [tex]\( x \rightarrow -\infty \)[/tex]:
[tex]\[ \lim_{{x \to -\infty}} f(x) = \lim_{{x \to -\infty}} (-5x^5) \][/tex]
Since [tex]\( x^5 \)[/tex] grows very large and negative as [tex]\( x \)[/tex] approaches [tex]\(-\infty \)[/tex]:
[tex]\[ -5x^5 \rightarrow +\infty \][/tex]
Therefore:
[tex]\[ \lim_{{x \to -\infty}} f(x) = +\infty \][/tex]
### Conclusion:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( +\infty \)[/tex].
- As [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \)[/tex] approaches [tex]\( -\infty \)[/tex].
Combining these conclusions:
- [tex]\( \textbf{As } x \to -\infty, f(x) \to +\infty \)[/tex].
- [tex]\( \textbf{As } x \to +\infty, f(x) \to -\infty \)[/tex].
