Rational functions Evaluate limT →∞ f(x) and limT →-∞ f(x)...

Rational functions Evaluate $\lim _{T \rightarrow \infty} f(x)$ and $\lim _{T \rightarrow-\infty} f(x)$ for the following rational functions. Then give the horizontal asymptote of $f$ (if any).
$$f(x)=\frac{3 x^{3}-7}{x^{4}+5 x^{2}}$$

Key Concepts

Limits at Infinity

Limits at infinity examine the behavior of functions as the independent variable grows without bound (positively or negatively). For rational functions, the dominant terms (those with the highest power) in the numerator and denominator determine these limits. This analysis is critical for understanding the end behavior of the function, leading to conclusions about convergence or divergence.

Rational Functions

Rational functions are expressions formed by the quotient of two polynomials. They are studied to understand how the ratio of the leading terms influences the function's behavior, particularly for large values of the variable. The comparison of the degrees of the numerator and denominator plays a crucial role in determining the function's limits and asymptotic behavior.

Horizontal Asymptotes

Horizontal asymptotes describe the horizontal line that the graph of a function approaches as the independent variable tends to infinity or negative infinity. In the context of rational functions, if the degree of the numerator is less than the degree of the denominator, the function approaches zero. When the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients, and if the numerator's degree is higher, there is no horizontal asymptote.

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