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

Welcome back, everyone. Find the limit. Limit as x approaches 3 from the left of 2 divided by square root of 2 X multiplied by X minus 3. We're given 4 answer choices A says 0, B infinity, C2, and D does not exist. So let's begin solving for the limit as X approaches 3 from the left using direct substitution. So we're going to attempt and substitute X equals 3 for every X that we have in our function. Now if we neglect 3 from the left and simply use 3, we're going to get 2 divided by square root of. 2 multiplied by 3, multiplied by 3 minus 3, right? And 3 minus 3 is going to give us 0. So scroot of 0 is 0, we get 2 divided by 0, which would be infinity. But now what we want to do is basically consider. That x approaches 3 from the left and we get 2 in the numerator and for the denominator we get square root of. Now 2 multiplied by 3 gives us 6, and now the most important part is 3 minus 3. Since we're approaching 3 from the left, we're going to say 3 from the left minus 3, and essentially this means that X is less than 3. And if we're subtracting 3 from a number that is less than 3, we're getting. An infinitely small negative number, so that's 0 from the left. This means that we get 2. Divided by Square root of An infinitely small negative number. And because square root of X is defined as long as X is greater than or equal to 0, we can say that this limit does not exist, right, because we cannot take square root of a negative number. And therefore, the correct answer to this problem corresponds to the answer choice D does not exist. Thank you for watching.

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1. Limits and Continuity

Finding Limits Algebraically

Calculus

1. Limits and Continuity

Finding Limits Algebraically

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2:49 minutes

Problem 2.4.29b

Textbook Question

Determine the following limits.

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

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Identify the limit expression: $ \lim_{x \to 2^{-}} \frac{1}{\sqrt{x(x-2)}} $.

Recognize that as $ x \to 2^{-} $, the expression $ x - 2 $ approaches 0 from the negative side, making the expression inside the square root approach 0.

Consider the behavior of $ \sqrt{x(x-2)} $ as $ x \to 2^{-} $. Since $ x $ is slightly less than 2, $ x(x-2) $ is negative, and the square root of a negative number is not real.

Conclude that the expression inside the square root becomes undefined in the real number system as $ x \to 2^{-} $.

Determine that the limit does not exist in the real number system due to the square root of a negative number.

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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 concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit as x approaches 2 from the left, which is denoted as x → 2⁻.

Square Root Function

The square root function, denoted as √x, is a mathematical function that returns the non-negative value whose square is x. In the context of limits, the behavior of the square root function can significantly affect the limit's value, especially when the argument of the square root approaches zero, as it can lead to undefined or infinite values.

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, either from the left (denoted as x → a⁻) or from the right (denoted as x → a⁺). This concept is crucial when dealing with functions that may behave differently on either side of a point, such as in this limit problem where x approaches 2 from the left.

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