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).


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

Verified step by step guidance

1. Step 1: Identify the degrees of the numerator and the denominator.
   The numerator is \(4x\), which is a polynomial of degree 1. The denominator is \(20x + 1\), also a polynomial of degree 1.
2. Step 2: Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients.
   The leading coefficient of the numerator is 4, and the leading coefficient of the denominator is 20.
3. Step 3: Calculate the horizontal asymptote by dividing the leading coefficients.
   \(\text{Horizontal Asymptote} = \frac{4}{20} = \frac{1}{5}\).
4. Step 4: Determine \(\lim_{x\rightarrow\infty}f(x)\).
   As \(x\) approaches infinity, the function \(f(x) = \frac{4x}{20x+1}\) approaches the horizontal asymptote \(\frac{1}{5}\).
5. Step 5: Determine \(\lim_{x\rightarrow-\infty}f(x)\).
   As \(x\) approaches negative infinity, the function \(f(x) = \frac{4x}{20x+1}\) also approaches the horizontal asymptote \(\frac{1}{5}\).

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.