Question

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{x}{|x|}$$

Short answer

The limit as \(x\) approaches \(+\infty\) is 1, the limit as \(x\) approaches \(-\infty\) is -1, and the horizontal asymptotes are at \(y = 1\) and \(y = -1\).

Step-by-step solution

Evaluate the Function for \(x>0\)

First, consider the case where \(x > 0\). For this, the absolute value of \(x\) is simply \(x\) itself. Hence, \(f(x) = \frac{x}{x} = 1\) for \(x > 0\). The limit then is \(\lim_{x \rightarrow \infty} f(x) = 1\).

Evaluate the Function for \(x

Next, consider the case where \(x < 0\). The absolute value of \(x\) for negative \(x\) numbers is \(-x\). So for \(x<0\), \(f(x) = \frac{x}{-x} = -1\). The limit then is \(\lim_{x \rightarrow -\infty} f(x) = -1\).

Identify horizontal asymptotes

A function \(f(x)\) has a horizontal asymptote \(y = L\) if \(\lim_{x \rightarrow \infty} f(x) = L\) or \(\lim_{x \rightarrow -\infty} f(x) = L\). Here, \(f(x)\) approaches 1 as \(x\) approaches \(+\infty\) and \(-1\) as \(x\) approaches \(-\infty\). Therefore, the horizontal asymptotes are \(y = 1\) and \(y = -1\).

Key concepts

Evaluating Limits

When trying to understand limits in calculus, consider it as an investigation into the behavior of a function as the input (usually denoted as \(x\)) approaches a specific value.

For the example \(f(x) = \frac{x}{|x|}\), evaluating the limit as \(x\) approaches infinity involves examining the function's values as \(x\) gets larger and larger. In the provided steps, we saw that for positive values of \(x\), the function simplifies to 1, because the absolute value of a positive number is the number itself. Hence, the limit as \(x\) approaches positive infinity is 1.

Similarly, for negative values of \(x\), the function simplifies to -1, since the absolute value of a negative number is its positive counterpart. Consequently, the limit as \(x\) approaches negative infinity is -1. This process of systematically evaluating a function to find its limit is essential for understanding the behavior of functions, especially as they approach infinite values.

Horizontal Asymptotes

Horizontal asymptotes represent the value that a function's output (\(y\)) will approach as the input (\(x\)) heads towards positive or negative infinity. They are like the horizon for sea travelers—no matter how far you go, the horizon is always out there, unattainable.

In this exercise, since \(f(x)\) approaches 1 as \(x\) approaches infinity and -1 as \(x\) approaches negative infinity, we identify two horizontal asymptotes: \(y = 1\) and \(y = -1\). These lines on a graph provide a valuable visual aid to understand the end behavior of functions. They also serve as a 'behavioral boundary' of sorts, which the function's values will always tend to but not necessarily ever touch.

Absolute Value

The concept of absolute value is crucial in evaluating functions that contain absolute value expressions. In basic terms, the absolute value of a number is its distance from zero on the number line, without considering the direction.

For the function \(f(x) = \frac{x}{|x|}\), the absolute value affects the function's output depending on whether \(x\) is positive or negative. If \(x\) is positive, the absolute value of \(x\) leaves it unchanged. Conversely, if \(x\) is negative, the absolute value converts it into a positive number. That's why, in the function \(f(x)\), the sign of \(x\) dictates whether the function's output is 1 or -1. Understanding absolute value is crucial for dissecting such piecewise functions that have different rules for different intervals of the input variable.