To determine the long-run behavior of rational functions, we need to analyze the degrees of the polynomials in the numerator and the denominator. Specifically, we compare the highest degree terms in both the numerator and the denominator.
For \(\frac{x^{2}+1}{x^{2}+2}\):
- The degrees of the numerator and the denominator are both 2.
- As \(x\) approaches infinity, the ratio of the leading coefficients will determine the behavior.
For \(\frac{x^{2}+1}{x^{3}+2}\):
- The degree of the numerator is 2, and the degree of the denominator is 3.
- As \(x\) approaches infinity, the denominator grows faster than the numerator, leading the function to approach 0.
For \(\frac{x^{3}+1}{x^{2}+2}\):
- The degree of the numerator is 3, and the degree of the denominator is 2.
- As \(x\) approaches infinity, the numerator grows faster than the denominator, leading the function to approach infinity.
To determine the long-run behavior of the function \(\frac{x^{2}+1}{x^{2}+2}\), we compare the degrees of the numerator and the denominator. Both the numerator and the denominator have the highest degree term \(x^2\).
As \(x \to \infty\), the function behaves like \(\frac{x^2}{x^2} = 1\).
For the function \(\frac{x^{2}+1}{x^{3}+2}\), the degree of the numerator is 2, and the degree of the denominator is 3.
As \(x \to \infty\), the denominator grows faster than the numerator, leading the function to approach 0.
For the function \(\frac{x^{3}+1}{x^{2}+2}\), the degree of the numerator is 3, and the degree of the denominator is 2.
As \(x \to \infty\), the numerator grows faster than the denominator, leading the function to approach \(\infty\).
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