Problem 5 - Calculus: Graphical, Numerical, Algebraic

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. $$f(x)=\frac{3 x+1}{|x|+2}$$

Step-by-Step Solution

Answer

The limits of the function \(f(x) = \frac{3 x + 1}{|x| + 2}\) as \(x\) approaches \(\infty\) and \(-\infty\) are 3 and -3 respectively. The function has horizontal asymptotes at \(y = 3\) and \(y = -3\).

Step 1: Break down the function

First we'll separate the function \(f(x)=\frac{3 x+1}{|x|+2}\) into two piecewise functions. Since \(|x|\) is \(x\) when \(x\) is positive and \(-x\) when \(x\) is negative, the function \(f(x)\) can be written as:\[f(x) = \begin{cases} \frac{3x+1}{x+2} & \text{if } x \geq 0 \\ \frac{3x+1}{-x+2} & \text{if } x < 0 \end{cases}\]

Step 2: Compute the limit as \(x\) approaches \(\infty\)

Now we can find \(\lim _{x \rightarrow \infty} f(x)\). This will be based on the first piece of the piecewise function, \(\frac{3x+1}{x+2}\). We can divide both the numerator and the denominator by \(x\) to simplify and get \(\frac{3 + \frac{1}{x}}{1 + \frac{2}{x}}\). As \(x \rightarrow \(\infty\)\), terms with \(\frac{1}{x}\) will approach 0 and we end up with \(\lim _{x \rightarrow \infty} \frac{3 + \frac{1}{x}}{1 + \frac{2}{x}} = \frac{3}{1} = 3\).

Step 3: Compute the limit as \(x\) approaches \(-\infty\)

To find \(\lim _{x \rightarrow -\infty} f(x)\) we would need to look at the second piece of the piecewise function, which is \(\frac{3x+1}{-x+2}\). Again, we can simplify by dividing both the numerator and the denominator by \(x\), to get \(\frac{3 - \frac{1}{x}}{-1 - \frac{2}{x}}\). As \(x \rightarrow -\infty\), we see that terms with \(\frac{1}{x}\) will approach 0, resulting in \(\lim _{x \rightarrow -\infty} \frac{3 - \frac{1}{x}}{-1 - \frac{2}{x}} = \frac{3}{-1} = -3\)

Step 4: Find the horizontal asymptotes

In this case, the horizontal asymptotes of the function are the limits at \(\infty\) and \(-\infty\). Therefore, the horizontal asymptotes of \(f(x)\) are \(y = 3\) and \(y = -3\).

Key Concepts

Limits at Infinity

Understanding the concept of limits at infinity is essential when studying calculus. When we say limits at infinity, we are describing the behavior of a function as the value of the independent variable (\( x \) in most cases) grows larger and larger, either positively or negatively.

In practical terms, we want to know what value the function is approaching as we move along the x-axis towards positive or negative infinity. For rational functions like the one in our exercise—\frac{3x+1}{x+2}\frac{3x+1}{x+2}—the limit at infinity usually depends on the degrees of the polynomial in the numerator and the denominator. If they are the same degree, the limit is the ratio of the coefficients of the highest degree terms, which in this case, gives us \( 3 \) as we approach \( \infty \).

For learners, a handy tip is to divide every term by the highest power of \( x \) found in the function to simplify your analysis. This helps to identify which terms become insignificant as \( x \) approaches infinity and makes the behavior of the function more evident.

Horizontal Asymptotes

The concept of horizontal asymptotes ties closely with limits at infinity. A horizontal asymptote is a horizontal line that the graph of a function approaches as \( x \) tends towards positive or negative infinity. However, it does not mean the function will reach these values; it simply gets closer and closer to the asymptote as it extends further along the x-axis.

In the given example, we have determined the limits of \( f(x) \) to be \( 3 \) as \( x \) approaches \infty\ and \( -3 \) as \( x \) approaches \(-\infty\). These limits imply that the horizontal asymptotes of \( f(x) \) are at \( y = 3 \) and \( y = -3 \) respectively. This means the graph of the function will get closer to these lines the further you go in either direction along the x-axis, but it won’t ever actually touch or cross these lines.

Piecewise Functions

A piecewise function is a function that is defined by multiple sub-functions, each applying to a certain interval of the main function’s domain. The function \( f(x) \) given in our exercise is split into two different expressions dependent on whether \( x \) is positive or negative due to the absolute value function in the denominator.

When handling piecewise functions, it becomes crucial to evaluate each sub-function individually to determine the function’s overall behavior. As in our example, the limit of \( f(x) \) as \( x \) approaches infinity requires us to only consider the sub-function that applies when \( x \) is positive. Likewise, for the limit as \( x \) approaches negative infinity, we use the sub-function valid for negative \( x \) values.

Remember, regardless of the expression for the function in separate intervals, the limit, if it exists, must be the same from either side of the point at which it is being evaluated for the graph to be continuous at that point.