Problem 49 - Single Variable Calculus

Question

In Exercises \(49-56\), find the horizontal and vertical asymptotes of the graph of the function. Do not sketch the graph. $$ f(x)=\frac{1}{x+2} $$

Step-by-Step Solution

Answer

The vertical asymptote is \(x = -2\), and the horizontal asymptote is \(y = 0\).

Step 1: Find the vertical asymptote

To find the vertical asymptote, we need to find the value of \(x\) where the denominator of the function equals zero. We have the denominator as \(x+2\). So, we set it equal to zero and solve for \(x\). \[ x + 2 = 0 \] Subtract 2 from both sides, and we get \(x = -2\). Thus, the vertical asymptote is the line \(x = -2\).

Step 2: Find the horizontal asymptotes

To find the horizontal asymptote(s), we need to analyze the behavior of the function as \(x\) approaches positive and negative infinity. If there is a limit to the function as \(x\) approaches either direction, then that limit becomes the horizontal asymptote. As \(x\) approaches positive infinity: \[ \lim_{x \to \infty} \frac{1}{x+2} \] As the value of x gets larger, the term \(x+2\) in the denominator also gets larger, making the fraction \(\frac{1}{x+2}\) approach 0. Thus, for positive infinity, the horizontal asymptote is \(y = 0\). As \(x\) approaches negative infinity: \[ \lim_{x \to -\infty} \frac{1}{x+2} \] As the value of x gets much smaller (more negative), the term \(x+2\) in the denominator still gets even larger but in the negative sense (more negative), making the fraction \(\frac{1}{x+2}\) approach 0 as well. Thus, for negative infinity, the horizontal asymptote is also \(y = 0\). So, for the function \(f(x) = \frac{1}{x+2}\), we have the following asymptotes: Vertical asymptote: \(x = -2\) Horizontal asymptote: \(y = 0\)

Key Concepts

Vertical Asymptote

Vertical asymptotes are special lines on a graph where the function does not actually reach a finite value, typically because the function explodes to infinity. You can think of these asymptotes like invisible walls that the function cannot cross. For example, if we have a function like

• \( f(x) = \frac{1}{x+2} \)

we find the vertical asymptote by setting the denominator equal to zero and solving for \( x \). This is because division by zero is undefined in mathematics. So:

• \( x + 2 = 0 \)
• Solve for \( x \):

\[ x = -2 \] This means our vertical asymptote is at \( x = -2 \). The graph of the function will approach this vertical line but never actually touch it. Why? As \( x \) gets closer and closer to -2, the function's value skyrockets toward either positive or negative infinity, depending on the direction from which \( x \) is approaching.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function gets closer and closer to as \( x \) goes to infinity or negative infinity. It doesn't mean the function will never cross this line, but as \( x \) becomes very large positively or negatively, it tends to flatten out and approach the asymptote.For our function \( f(x) = \frac{1}{x+2} \), let's see how it behaves:**Behavior at Positive Infinity**

• As \( x \to \infty \), the denominator \( x+2 \) becomes very large.
• This makes the whole fraction \( \frac{1}{x+2} \) approach \( 0 \).

Thus, the horizontal asymptote is \( y = 0 \).**Behavior at Negative Infinity**

• Similarly, as \( x \to -\infty \), \( x+2 \) also becomes very large in magnitude but negative.
• Nonetheless, the function's value continues to approach \( 0 \).

Once again, the horizontal asymptote is \( y = 0 \). This tells us that as \( x \) goes to extreme ends, both positively and negatively, the function flattens out near the \( x \)-axis.

Limits in Calculus

Limits in calculus are a fundamental concept that help describe how a function behaves as it approaches a specific point or extends toward infinity. This is crucial for finding both vertical and horizontal asymptotes.**Understanding Limits**

• A limit considers the value that a function approaches as the input (\( x \)) nears a specific point.
• For vertical asymptotes, we use limits to assess what happens to \( f(x) \) as \( x \) approaches a critical value where the function is undefined.
• This helps confirm if the function tends toward infinity, indicating an asymptote.

**Using Limits for Horizontal Asymptotes**

• To find horizontal asymptotes, we calculate the limit of the function as \( x \to \infty \) and \( x \to -\infty \).
• If the function approaches a specific finite value \( L \) in these cases, then \( y = L \) is a horizontal asymptote.

Thus, limits are an indispensable tool in understanding the behavior of graphs, giving us insight into asymptotes' locations and nature, as illustrated in the function \( f(x) = \frac{1}{x+2} \).