To determine the end behavior of the rational function [tex]\( f(x) = \frac{x+1}{5x^3+1} \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches infinity ([tex]\( \infty \)[/tex]) and negative infinity ([tex]\( -\infty \)[/tex]).
First, let's determine the degrees of the numerator and the denominator:
- The numerator is [tex]\( x + 1 \)[/tex]. The highest power of [tex]\( x \)[/tex] in the numerator is [tex]\( x^1 \)[/tex], so the degree of the numerator is 1.
- The denominator is [tex]\( 5x^3 + 1 \)[/tex]. The highest power of [tex]\( x \)[/tex] in the denominator is [tex]\( x^3 \)[/tex], so the degree of the denominator is 3.
The behavior of rational functions as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex] is largely determined by the degrees of the numerator and the denominator.
Since the degree of the numerator (1) is less than the degree of the denominator (3), we can infer that the end behavior of [tex]\( f(x) \)[/tex] will approach 0.
We can further simplify and confirm this by focusing on the leading terms of the numerator and the denominator:
[tex]\[ f(x) \approx \frac{x}{5x^3} = \frac{1}{5x^2} \][/tex]
As [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex]:
- The term [tex]\( \frac{1}{5x^2} \)[/tex] approaches 0 because [tex]\( 5x^2 \)[/tex] grows without bound, making the fraction's value get smaller and smaller.
So, as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex]:
- Similarly, the term [tex]\( \frac{1}{5x^2} \)[/tex] also approaches 0 because [tex]\( 5x^2 \)[/tex] is always positive (since [tex]\( x^2 \)[/tex] is positive regardless of whether [tex]\( x \)[/tex] is positive or negative), and it also grows without bound.
So, as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
Therefore, the end behavior of the function [tex]\( f(x) \)[/tex] is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to 0 \)[/tex].
In conclusion, the function [tex]\( f(x) = \frac{x+1}{5x^3+1} \)[/tex] behaves such that:
[tex]\[ \boxed{(0, 0)} \][/tex]
