Textbook Question
Evaluate each limit and justify your answer.
lim x→1 (x+5x / x+2)^4
1. Limits and Continuity
Finding Limits Algebraically
1:42 minutesProblem 2.12a
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.
a. lim x→−2^+ f(x)
Verified step by step guidance1
Step 1: Understand the function and its components. The function is \( f(x) = \frac{e^{-x}}{x(x+2)^2} \). It is a rational function with an exponential term in the numerator and a polynomial in the denominator.
Step 2: Identify the point of interest for the limit, which is \( x \to -2^+ \). This means we are approaching \( x = -2 \) from the right side.
Step 3: Analyze the behavior of the denominator as \( x \to -2^+ \). The term \((x+2)^2\) in the denominator approaches zero, which suggests a vertical asymptote at \( x = -2 \).
Step 4: Consider the behavior of the numerator, \( e^{-x} \), as \( x \to -2^+ \). Since \( e^{-x} \) is continuous and positive for all real \( x \), it does not approach zero or infinity.
Step 5: Use a graphing utility to visualize the function \( f(x) \) and observe the behavior as \( x \to -2^+ \). The graph will help confirm whether the limit approaches positive or negative infinity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 -2 from the right (denoted as -2^+) involves analyzing the behavior of the function f(x) near that point, which can reveal important characteristics such as continuity and potential asymptotes.
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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 key features such as intercepts, asymptotes, and the overall shape of the graph, aiding in the limit evaluation.
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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 -2, understanding whether the function approaches a finite value, diverges to infinity, or oscillates is crucial for determining the limit. This behavior can often be inferred from the graph and the function's algebraic form.
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