Determine ${\lim }_{x\to \infty }f\left(x\right)$ and ${\lim }_{x\to -\infty }f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).


$f\left(x\right)=\frac{4x}{20x+1}$

Key Concepts

Limits at Infinity

Limits at infinity describe the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine whether they approach a specific value, diverge, or oscillate. Evaluating these limits often involves simplifying the function to its leading terms, especially in rational functions.

Horizontal Asymptotes

Horizontal asymptotes indicate the value that a function approaches as the input approaches infinity or negative infinity. They are determined by the limits at infinity and provide insight into the long-term behavior of the function. If a function has a horizontal asymptote, it means that as x becomes very large or very small, the function stabilizes around a particular value.

Rational Functions

Rational functions are ratios of two polynomials, expressed in the form f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the degrees of the polynomials in the numerator and denominator is essential for analyzing limits and asymptotic behavior. The leading coefficients and degrees dictate the limits at infinity and the presence of horizontal asymptotes.