Problem 41 - Calculus with Concepts in Calculus

Question

Find all vertical and horizontal asymptotes of the graph of \(f\). You may wish to use a graphics calculator to assist you. $$ f(x)=x e^{1 / x} $$

Step-by-Step Solution

Answer

Function has no vertical or horizontal asymptotes.

Step 1: Understanding Vertical Asymptotes

Vertical asymptotes occur when the function approaches infinity as the input approaches a specific value. For the function \( f(x) = x e^{1/x} \), consider where \( e^{1/x} \) may become undefined or where division by zero is possible. This happens as \( x \) approaches zero. We evaluate the limit \( \lim_{{x \to 0^+}} f(x) \) and \( \lim_{{x \to 0^-}} f(x) \) to determine the behavior. In these cases, the exponential term does not cause an asymptote; instead, we find a discontinuity influenced by \( x \). Calculate these limits to confirm.

Step 2: Calculating Horizontal Asymptotes

Horizontal asymptotes are determined as \( x \) approaches infinity or negative infinity. Evaluate \( \lim_{{x \to \infty}} f(x) \) and \( \lim_{{x \to -\infty}} f(x) \). As \( x \) becomes very large, \( e^{1/x} \) approaches 1, therefore \( f(x) \) approaches \( x \cdot 1 = x \), indicating that there is no horizontal asymptote as \( x \to \infty \). As \( x \to -\infty \), similarly the behavior is complex without straightforward horizontal lines, suggesting asymptotic behavior aligns with \( y = x \). Re-evaluate specific bounds if needed.

Step 3: Limit Calculation for Asymptotic Behavior

Calculate \( \lim_{{x \to 0}} f(x) \). Consider \( \lim_{{x \to 0^+}} \) and \( \lim_{{x \to 0^-}} \). \( f(x) = x e^{1/x} \) shows complicated behavior as \( x \to 0 \). \( e^{1/x} \to \infty \) as \( x \to 0^+ \), and \( \to 0 \) as \( x \to 0^- \). However, multiplying by \( x \), the overall product \( f(x) \to 0 \). Thus, there is technically no vertical asymptote despite the infinite growth/shrink.

Step 4: Conclusion

No vertical asymptotes occur since \( \lim_{{x \to 0}} f(x) \to 0 \). No horizontal asymptotes occur since the function tends not towards a horizontal limit, with behavior \( y = x \) aligning closer than a typical linear horizontal asymptote. Consider graphical analysis to confirm border-like behaviors.

Key Concepts

Vertical Asymptotes

Vertical asymptotes are fascinating elements in mathematical analysis where a function seems to shoot off towards infinity at a specific value of its input. This occurrence typically signals a point of division by zero but is not always the case here. Let's consider the function

• \( f(x) = x e^{1/x} \).

For vertical asymptotes, we look at values of \(x\) that cause division by zero or make the function undefined. Here, this critical point is around \( x = 0 \). However, as explained in the solution, we need to consider the limits

• \( \lim_{{x \to 0^+}} f(x) \) and \( \lim_{{x \to 0^-}} f(x) \).

For this function, the behavior is quite nuanced. Although the exponential portion \( e^{1/x} \) can grow or shrink infinitely as \(x\) nears zero, multiplying this with \(x\) results in a product that tends towards 0 in the overall limit. Consequently, there's no vertical asymptote, because the output doesn't go to infinity near \( x = 0 \), but instead stabilizes around 0.

Horizontal Asymptotes

A horizontal asymptote is a line which the graph of a function gets infinitely close to as \(x\) goes to either positive or negative infinity. To find horizontal asymptotes for

• \( f(x) = x e^{1/x} \),

we look at two limits:

• \( \lim_{{x \to \infty}} f(x) \)
• \( \lim_{{x \to -\infty}} f(x) \).

For large positive \(x\), the expression \( e^{1/x} \approx 1\), so our function essentially behaves like \(x\). This means it doesn't trend towards a horizontal value, instead it heads further into infinity along the line \(y = x\). Similarly, when \(x\) approaches negative infinity, the analysis remains similar, resulting in no clear horizontal boundary. The function loosely follows an asymptotic path along \(y = x\), suggesting its behavior doesn't suit typical horizontal asymptote properties. Graphing may visually confirm this complex interplay.

Limit Calculation

Limit calculation plays a crucial role in understanding asymptotic behaviors of functions. With

• \( f(x) = x e^{1/x} \),

calculating limits provides insight into vertical and horizontal asymptotes. We focus on limits at crucial points: near 0, and at infinity.

• Near \( x = 0 \): the limits \( \lim_{{x \to 0^+}} f(x) \) and \( \lim_{{x \to 0^-}} f(x) \) reveal that the function tends towards 0, rather than infinity, despite varied directions in growth from \( e^{1/x} \). Multiply by \( x \) balances this to a stabilizing 0.
• For \( x \to \infty \) or \( x \to -\infty \), checking \( \lim_{{x \to \infty}} f(x) \) or \( \lim_{{x \to -\infty}} f(x) \), highlights the strategy of simplification. Approximating \( e^{1/x} \approx 1\) allows us to see the behavior aligns with \(y = x\), absent of strict horizontal limits.

Limit calculations clarify that asymptotic paths can streamline towards linear correlations rather than definite horizontal or vertical confines.