Problem 58 - Calculus and its applications

Question

Graph each function and then find the specified limits. When necessary, state that the limit does not exist. $$ f(x)=\frac{1}{x}+3 ; \quad \text { find } \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow 0} f(x). $$

Step-by-Step Solution

Answer

\( \lim_{x \rightarrow \infty} f(x) = 3 \); \( \lim_{x \rightarrow 0} f(x) \) does not exist.

Step 1: Understand the Function

The function given is \( f(x) = \frac{1}{x} + 3 \). This function consists of two parts: the fraction \( \frac{1}{x} \) and a constant 3. The fraction will modify the behavior of the graph, especially near 0, and the entire function will be shifted up by 3 due to the constant.

Step 2: Analyze the Asymptotic Behavior as \( x \rightarrow \infty \)

As \( x \rightarrow \infty \), \( \frac{1}{x} \rightarrow 0 \). Therefore, the function behaves as \( f(x) \approx 3 \). Thus, \( \lim_{x \rightarrow \infty} f(x) = 3 \).

Step 3: Analyze the Asymptotic Behavior as \( x \rightarrow 0 \)

As \( x \rightarrow 0 \), the behavior of \( \frac{1}{x} \) depends on the direction of approach. If \( x \rightarrow 0^+ \), \( \frac{1}{x} \rightarrow +\infty \), and if \( x \rightarrow 0^- \), \( \frac{1}{x} \rightarrow -\infty \). Thus, \( f(x) \rightarrow +\infty \) as \( x \rightarrow 0^+ \) and \( f(x) \rightarrow -\infty \) as \( x \rightarrow 0^- \). The two-sided limit \( \lim_{x \rightarrow 0} f(x) \) does not exist.

Step 4: Graph the Function

Sketch the graph of \( f(x) = \frac{1}{x} + 3 \). The graph will have a horizontal asymptote at \( y = 3 \) as \( x \rightarrow \pm \infty \), and a vertical asymptote at \( x = 0 \). The curve approaches 3 as \( x \rightarrow \infty \) or \( x \rightarrow -\infty \), and shoots towards \( \pm \infty \) as \( x \rightarrow 0 \).

Key Concepts

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as the input approaches a particular value or infinity. It's like watching a friend carefully sneak up to a finish line but never quite crossing it. Mathematically, this involves looking at how the function behaves as we move towards certain points on the x-axis, such as 0 or infinity.
In our example function, \( f(x)= \frac{1}{x} + 3 \), we are interested in what happens when \( x \) approaches 0 and when \( x \) approaches infinity.

• As \( x \to \infty \), \( \frac{1}{x} \to 0 \). The entire function then gets close to 3, establishing our horizontal asymptote.
• As \( x \to 0 \), things get more exciting. From the positive side (\( x \to 0^+ \)), \( \frac{1}{x} \to \infty \), making the function shoot upwards. From the negative side (\( x \to 0^- \)), \( \frac{1}{x} \to -\infty \), causing it to dive.

This split behavior around 0 gives rise to a vertical asymptote, as the function can't actually handle \( x=0 \). By looking at these extremes, we see how a function stretches and bends but never quite touches these magical lines.

Horizontal Asymptote

A horizontal asymptote tells us what value the function is approaching, as the input gets very large or very small. It represents a stable state, a kind of balance the function approaches but rarely achieves completely.
For functions like \( f(x)= \frac{1}{x} + 3 \), you check the behavior as \( x \to \pm \infty \).

• As \( x \to \infty \), \( \frac{1}{x} \) becomes negligible, and the function settles close to 3. So, \( f(x) \approx 3 \).
• Similarly, as \( x \to -\infty \), the behavior is the same: \( f(x) \approx 3 \).

That's why there's a horizontal asymptote at \( y=3 \).
This line is like the highway that the function travels alongside but never merges onto, no matter how far you drive.

Vertical Asymptote

Vertical asymptotes are lines that indicate where a function experiences infinite "growth" or "fall," creating a dramatic effect on the graph. It's where the function takes a wild turn, unable to reach a real number output, essentially becoming undefined.
In our exercise, the function \( f(x) = \frac{1}{x} + 3 \) has a vertical asymptote at \( x=0 \).

• As \( x \to 0^+ \), or from the right, the function goes way up because \( \frac{1}{x} \to \infty \).
• Conversely, as \( x \to 0^- \), or from the left, it dives downward with \( \frac{1}{x} \to -\infty \).

Here, the function doesn't settle in towards a horizontal line; instead, it shoots vertically, leaving a gap at \( x=0 \).
This vertical line at \( x=0 \) on the graph shows discontinuity and challenges students to understand why the function behaves so uniquely in these spots. By inspecting the parts of the function around this line, you can grasp why the graph "blows up" at specific points.