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.


d. lim x→0^+ f(x)

Verified step by step guidance

1. Identify the function: \( f(x) = \frac{e^{-x}}{x(x+2)^2} \).
2. Focus on the limit as \( x \to 0^+ \), which means approaching 0 from the right.
3. Consider the behavior of each part of the function as \( x \to 0^+ \): \( e^{-x} \to 1 \), \( x \to 0^+ \), and \( (x+2)^2 \to 4 \).
4. Analyze the expression \( \frac{1}{x} \) as \( x \to 0^+ \), which tends to infinity.
5. Conclude that the limit \( \lim_{x \to 0^+} f(x) \) depends on the behavior of \( \frac{1}{x} \) and the other terms.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. In this case, evaluating the limit as x approaches 0 from the right (0+) is crucial for determining the function's behavior near that point.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a * e^(bx), where e is Euler's number (approximately 2.71828). In the given function f(x) = e^(-x) / (x(x+2)^2), the exponential component e^(-x) influences the function's growth or decay as x changes. Understanding how exponential functions behave as x approaches certain values is essential for analyzing limits.

Graphing Rational Functions

Rational functions are ratios of polynomials, and their graphs can reveal important information about their limits and asymptotic behavior. The function f(x) = e^(-x) / (x(x+2)^2) is a rational function, and graphing it allows for visualizing its behavior near critical points, such as x = 0. Analyzing the graph helps in determining the limit as x approaches 0 from the right, as well as identifying any vertical or horizontal asymptotes.