To determine the end behavior of the graph of [tex]\( f(x) = x^3 (x+3) (-5x+1) \)[/tex] using limits, we'll analyze the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\( +\infty \)[/tex].
1. Examining [tex]\( f(x) \)[/tex] as [tex]\( x \to -\infty \)[/tex]:
- At a large negative [tex]\( x \)[/tex], [tex]\( x \)[/tex] is a large negative number, [tex]\( x + 3 \)[/tex] is also large and negative, and [tex]\( -5x + 1 \)[/tex] becomes positive since [tex]\( -5 \times \text{negative\_large\_number} \)[/tex] will be positive.
- Therefore, multiplying these together [tex]\(( x^3 (x+3) (-5x+1) )\)[/tex], the product of two negative numbers (which is positive) and one positive number is positive. Since the term with the highest degree, [tex]\( x^5 \)[/tex], has a negative coefficient, the overall function will be highly negative.
Hence, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
2. Examining [tex]\( f(x) \)[/tex] as [tex]\( x \to +\infty \)[/tex]:
- At a large positive [tex]\( x \)[/tex], [tex]\( x \)[/tex] is a large positive number, [tex]\( x + 3 \)[/tex] is also a large positive number, and [tex]\( -5x + 1 \)[/tex] becomes negative since [tex]\( -5 \times \text{positive\_large\_number} \)[/tex] will be a large negative.
- Therefore, multiplying these together [tex]\(( x^3 (x+3) (-5x+1))\)[/tex], the product of three positive numbers (two in the significant domain of negative) and one negative number is negative.
- Similarly, since the term with the highest degree, [tex]\( x^5 \)[/tex], has a negative coefficient, the function will be highly negative.
Hence, as [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
To summarize:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to +\infty \)[/tex].
- As [tex]\( x \to +\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
So, the function exhibits the following end behavior:
[tex]\[ \text{As } x \rightarrow -\infty, f(x) \rightarrow -\infty \][/tex]
[tex]\[ \text{As } x \rightarrow +\infty, f(x) \rightarrow -\infty \][/tex]
[tex]\[ \text{As } x \rightarrow -\infty, f(x) \rightarrow +\infty \][/tex]
[tex]\[ \text{As } x \rightarrow +\infty, f(x) \rightarrow \infty \][/tex]
[tex]\[ \text{As } x \rightarrow -\infty, f(x) \rightarrow +\infty \][/tex]
[tex]\[ \text{As } x \rightarrow +\infty, f(x) \rightarrow \infty \][/tex]
