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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.12a
Chapter 2, Problem 2.12a

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 guidance
1
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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One-Sided Limits

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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Related Practice
Textbook Question

Complete the following sentences in terms of a limit.


a. A function is continuous from the left at a if _____.

Textbook Question

The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>

lim x→−2^− h(x)

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.


a. The graph of a function can never cross one of its horizontal asymptotes.

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Textbook Question

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1^− f(x)

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Textbook Question

The hyperbolic cosine function, denoted cosh(x)\(\cosh\]\left\)(x\(\right\)), is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as cosh(x)=ex+e−x2\(\cosh\]\left\)(x\(\right\))=\(\frac{e^{x}\)+e^{-x}}{2}.


a. Determine its end behavior by analyzing limx→∞cosh(x){\(\displaystyle\[\lim\)_{x\(\to\]\infty\)}{\(\cosh\)(x)}} and limx→−∞cosh(x){\(\displaystyle\]\lim\)_{x\(\to\)-\(\infty\)}{\(\cosh\)(x)}}.

Textbook Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x2 − 9)/(x(x−3))