Graph the function f(x)=e^−x / x(x+2)^2 using a graphing utility. (Experiment with your choice of a graphing window.) Use your graph to determine the following limits.


c. lim x→0^− f(x)

Verified step by step guidance

1. Step 1: Understand the function. The function given is \( f(x) = \frac{e^{-x}}{x(x+2)^2} \). We need to analyze this function as \( x \) approaches 0 from the left (\( x \to 0^- \)).
2. Step 2: Consider the behavior of the function as \( x \to 0^- \). Note that the denominator \( x(x+2)^2 \) will approach 0, which suggests a potential vertical asymptote or undefined behavior at \( x = 0 \).
3. Step 3: Analyze the numerator and denominator separately. The numerator \( e^{-x} \) approaches \( e^0 = 1 \) as \( x \to 0^- \). The denominator \( x(x+2)^2 \) approaches 0, but since \( x \to 0^- \), \( x \) is negative, making the denominator negative.
4. Step 4: Combine the behavior of the numerator and denominator. Since the numerator approaches 1 and the denominator approaches a small negative value, the overall function \( f(x) \) will approach negative infinity as \( x \to 0^- \).
5. Step 5: Use a graphing utility to confirm the behavior. Graph \( f(x) = \frac{e^{-x}}{x(x+2)^2} \) and observe the behavior as \( x \to 0^- \). The graph should show the function approaching negative infinity, confirming the limit.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, evaluating the limit as x approaches 0 from the left (denoted as x→0^−) involves analyzing the behavior of the function f(x) near that point, which can reveal important characteristics about continuity and behavior of the function.

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate system to visualize its behavior. For the function f(x)=e^−x / x(x+2)^2, using a graphing utility allows for experimentation with different viewing windows, which can help identify asymptotic behavior, intercepts, and the overall shape of the graph, aiding in the understanding of limits.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as it approaches a certain point or infinity. In the case of f(x) as x approaches 0 from the left, understanding whether the function approaches a finite value, diverges to infinity, or approaches negative infinity is crucial for determining the limit. This behavior can often be inferred from the graph and the function's algebraic form.