Textbook Question
Evaluate each limit and justify your answer.
lim x→0 (x / √16x+1-1)^1/3
1. Limits and Continuity
Finding Limits Algebraically
1:45 minutesProblem 12c
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.
c. lim x→0^− f(x)
Verified step by step guidance1
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 (\( x \to 0^- \)).
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 \( x = 0 \).
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.
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 \( x \to 0^- \).
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.
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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 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.
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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 asymptotic behavior, intercepts, and the overall shape of the graph, aiding in the understanding of limits.
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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 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.
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