Determine the following limits.a. lim x → 2 + 1 x ( x − 2 ) {\displaystyle\lim_{x\to2^{+}}}\frac{1}{\sqrt{x\left(x-2\right)}}

Welcome back, everyone. In this problem, we want to find the limit as X approaches 4 from the right of 3 divided by the square root of X multiplied by X minus 4. A says it's negative infinity, B infinity, C 0, and D says the limit does not exist. Now here in this problem we can tell that the denominator of the fraction, the square root of x multiplied by x minus 4, approaches 0 as x approaches 4 from the right. In other words, we know that the square root of X multiplied by x minus 4 approaches 0 as X approaches 4 from the right. And it approaches 0 in such a way that the square root is always defined and positive because X is greater than 4. This means that the denominator of the fraction becomes very small and positive, making the fraction itself grow without bones. Therefore, our limit. As X approaches 4 from the right. Of function, OK. It is going to be equal to 3 divided by 0 coming from the right and that is going to be equal to infinity. Therefore, the limit is infinity. B is the correct answer. Thanks for watching everyone. I hope this video helped.

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1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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Problem 2.4.29a

Textbook Question

Determine the following limits.

a. $\lim_{x \to 2^{+}} \frac{1}{\sqrt{x (x - 2)}}$

Verified step by step guidance

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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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 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⁺.

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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Master Finding Limits by Direct Substitution with a bite sized video explanation from Patrick

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Determine the following limits.a. limx→2+1x(x−2){\displaystyle\lim_{x\to2^{+}}}\frac{1}{\sqrt{x\left(x-2\right)}}

Key Concepts

Limits

One-Sided Limits

Square Root Function

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Determine the following limits.

a. $\lim_{x \to 2^{+}} \frac{1}{\sqrt{x (x - 2)}}$