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→^3− h(x)
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1. Limits and Continuity
Finding Limits Algebraically
1:31 minutesProblem 2.4.29a
Textbook Question
Determine the following limits.
a.
Verified step by step guidance1
Step 1: Identify the limit expression: \( \lim_{x \to 2^{+}} \frac{1}{\sqrt{x(x-2)}} \). This is a one-sided limit as \( x \) approaches 2 from the right.
Step 2: Analyze the behavior of the expression as \( x \to 2^{+} \). Note that \( x - 2 \) approaches 0 from the positive side, making the expression inside the square root approach 0.
Step 3: Consider the expression \( \sqrt{x(x-2)} \). As \( x \to 2^{+} \), \( x \) is slightly greater than 2, so \( x(x-2) \) is a small positive number, and \( \sqrt{x(x-2)} \) is also a small positive number.
Step 4: Evaluate the behavior of the entire fraction \( \frac{1}{\sqrt{x(x-2)}} \). As \( \sqrt{x(x-2)} \) approaches 0 from the positive side, the fraction approaches infinity.
Step 5: Conclude that the limit is \( +\infty \) as \( x \to 2^{+} \), since the denominator approaches 0 from the positive side, causing the fraction to grow without bound.
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Video duration:
1mKey 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 where they may not be defined. In this question, we are specifically looking at the limit as x approaches 2 from the right, which is denoted as x → 2⁺.
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One-Sided Limits
One-Sided Limits
One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only. The limit from the right (denoted as x → c⁺) considers values greater than c, while the limit from the left (x → c⁻) considers values less than c. This distinction is crucial in this problem, as we are evaluating the limit as x approaches 2 from the right.
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Square Root Function
The square root function, denoted as √x, is defined for non-negative values of x and is important in this limit problem. The expression under the square root, x(x - 2), must be non-negative for the limit to be defined. Understanding the behavior of the square root function near critical points, such as where the argument becomes zero, is essential for evaluating the limit correctly.
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