Problem 37 - Calculus: Early Transcendentals

Question

Horizontal asymptotes Determine \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) for the following functions. Then give the horizontal asymptotes of \(f(\text {if any})\). $$f(x)=\frac{4 x}{20 x+1}$$

Step-by-Step Solution

Answer

Answer: The horizontal asymptote of the function is at \(y = \frac{1}{5}\).

Step 1: Identify the function

The function given is: $$f(x) = \frac{4x}{20x + 1}$$

Step 2: Calculate the limit as x approaches positive infinity

We need to find the following limit: $$\lim_{x \to \infty} \frac{4x}{20x + 1}$$ Divide the numerator and denominator by the highest power of x (in this case, x^1): $$\lim_{x \to \infty} \frac{\frac{4x}{x}}{\frac{20x + 1}{x}} = \lim_{x \to \infty} \frac{4}{20+\frac{1}{x}}$$ As x approaches infinity, the term \(\frac{1}{x}\) approaches 0: $$\lim_{x \to \infty} \frac{4}{20+0} = \frac{4}{20} = \frac{1}{5}$$

Step 3: Calculate the limit as x approaches negative infinity

We need to find the following limit: $$\lim_{x \to -\infty} \frac{4x}{20x + 1}$$ Using the same approach as in step 2, divide the numerator and denominator by the highest power of x (x^1): $$\lim_{x \to -\infty} \frac{\frac{4x}{x}}{\frac{20x + 1}{x}} = \lim_{x \to -\infty} \frac{4}{20+\frac{1}{x}}$$ As x approaches negative infinity, the term \(\frac{1}{x}\) again approaches 0: $$\lim_{x \to -\infty} \frac{4}{20+0} = \frac{4}{20} = \frac{1}{5}$$

Step 4: Determine the horizontal asymptotes

We found both limits to be the same value, \(\frac{1}{5}\). Therefore, there is a horizontal asymptote at $$y = \frac{1}{5}$$

Key Concepts

Limits at Infinity

Grasping the concept of limits at infinity is crucial to understanding the behavior of functions as the input values become very large, either positively or negatively. In our exercise, we evaluate the limits of the rational function as the variable x approaches positive and negative infinity. By dividing both the numerator and the denominator by x, we obtain a simpler expression that allows us to easily see what happens when x becomes infinitely large. As x grows, any term in the function that involves x in the denominator vanishes because it becomes an insignificantly small fraction.

Therefore, in the given exercise, as we let x approach infinity or negative infinity, the \(1/x\) term in the denominator reduces to zero, leaving us with a constant value. This observation is vital because it reveals that the function will approach a certain value (a horizontal line) as x gets extremely large or extremely small, which is essentially the horizontal asymptote of the function.

Asymptotic Behavior

The idea of asymptotic behavior can sometimes intimidate students, but it's essentially about understanding how a function behaves as it moves towards an infinite input value. The term 'asymptote' refers to a line that a graph of a function approaches but never actually touches.

In the given problem, the horizontal line y = 1/5 acts as a horizontal asymptote for the rational function. Both when approaching positive and negative infinity, our function f(x) will get arbitrarily close to this line but will not cross it. This property is key in sketching accurate graphs and predicting the function's long-term behavior. Horizontal asymptotes indicate the value that the function outputs will stabilize to, as the inputs become extremely large in magnitude.

Understanding how to determine these horizontal asymptotes is essential as it not only aids in graphing but also gives insight into the limits to growth or decay within real-world scenarios modeled by functions.

Rational Functions

A rational function is characterized by the division of two polynomials. The function from the exercise, \(f(x) = \frac{4x}{20x + 1}\), is a prime example, with its numerator and denominator being first-degree polynomials. The behavior of rational functions as x approaches infinity can often be determined by observing the degrees of these polynomials.

If the polynomial in the numerator has a lower degree than the denominator (as is the case in our function where both are of the same degree), the function has a horizontal asymptote at y = 0. In the case where they are of the same degree, as x tends to infinity, the function approaches the ratio of the leading coefficients, which is \( \frac{1}{5} \) in the given problem due to the coefficients 4 and 20 of x in the numerator and denominator respectively.

The importance of recognizing rational functions lies in their predictability. Many physical, economical, and biological systems can be modeled by such functions, where understanding their asymptotic behavior can lead to perceptive insights about the system's limitations and behaviors at extreme values.