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)
1. Limits and Continuity
Finding Limits Algebraically
1:35 minutesProblem 22b
Textbook Question
Determine the following limits.
b. lim x→3^− 2/(x − 3)^3
Verified step by step guidance1
Step 1: Identify the type of limit. This is a one-sided limit as \( x \) approaches 3 from the left (denoted by \( x \to 3^- \)).
Step 2: Recognize that the expression \( \frac{2}{(x - 3)^3} \) involves a power of \( (x - 3) \) in the denominator, which will approach zero as \( x \to 3^- \).
Step 3: Analyze the behavior of \( (x - 3)^3 \) as \( x \to 3^- \). Since \( x \) is approaching 3 from the left, \( x - 3 \) is negative, and thus \( (x - 3)^3 \) will also be negative and approach zero.
Step 4: Consider the sign of the entire expression \( \frac{2}{(x - 3)^3} \). Since the numerator is positive (2) and the denominator is negative and approaching zero, the overall expression will approach negative infinity.
Step 5: Conclude that the limit is \( -\infty \) as \( x \to 3^- \). This indicates that the function \( \frac{2}{(x - 3)^3} \) decreases without bound as \( x \) approaches 3 from the left.
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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 how functions behave near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit as x approaches 3 from the left (denoted as x→3^−).
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One-Sided Limits
One-Sided Limits
One-sided limits refer to the limits of a function as the input approaches a specific value from one side only, either the left or the right. The notation x→3^− indicates that we are considering the limit as x approaches 3 from values less than 3. This is crucial for analyzing functions that may behave differently from each side of a point.
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Behavior of Rational Functions
Rational functions are ratios of polynomials, and their limits can often be determined by analyzing the behavior of the numerator and denominator as the variable approaches a certain value. In this case, as x approaches 3, the denominator (x - 3)^3 approaches zero, which can lead to infinite limits or undefined behavior, depending on the sign of the numerator.
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