Textbook Question
Evaluate each limit and justify your answer.
lim x→5 ln 6(√x^2−16−3) / 5x−25
1. Limits and Continuity
Finding Limits Algebraically
1:32 minutesProblem 12d
Textbook Question
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 guidance1
Identify the function: \( f(x) = \frac{e^{-x}}{x(x+2)^2} \).
Focus on the limit as \( x \to 0^+ \), which means approaching 0 from the right.
Consider the behavior of each part of the function as \( x \to 0^+ \): \( e^{-x} \to 1 \), \( x \to 0^+ \), and \( (x+2)^2 \to 4 \).
Analyze the expression \( \frac{1}{x} \) as \( x \to 0^+ \), which tends to infinity.
Conclude that the limit \( \lim_{x \to 0^+} f(x) \) depends on the behavior of \( \frac{1}{x} \) and the other terms.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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