Determine the following limits.b. lim x → 3 x − 3 x 4 − 9 x 2 {\displaystyle\lim_{x\to3}}\frac{x-3}{x^4-9x^2}

Welcome back, everyone. Find the limit. Limit as X approaches 4 of 2 X minus 8 divided by X cubed minus 16 X. And we're given 4 answer choices A 1 divided by 32, B 1 divided by 16, C 1 divided by 64, and D 1 divided by 4. So let's analyze the given limit. It says limit as x approaches 4 of 2 X minus 8. Divided by X cubed minus 16 X. If we try the direct substitution, well, we're going to get 0 in the numerator, right? X equals 4, to multiply it by 4 minus 8 gives us 0, and in the denominator, 4 cubed minus 16 multiplied by 4, this gives us 0 as well. And this is an indeterminate form, so we can simply perform factorization for the numerator and the denominator. For the numerator we can factor out 2, and this leaves us with x minus 4. For the denominator, we have X cubed minus 16 X. So first of all, we can factor out X, which leaves us with X2 minus 16. And from here we can apply the formula for the difference of two squares. Because we have X2 minus 42 in the denominator. So this gives us 2 multiplied by X minus 4 in the numerator, and in the denominator we get X multiplied by X minus 4, multiplied by X + 4 based on the formula. And from here we can see that we have a hole. We can cancel out the common factor X minus 4 on both sides of the given fraction. And the limit becomes limit as X approaches 4 of 2 divided by X multiplied by X plus 4. And once again, let's apply the direct substitution, which gives us 2 divided by 4 multiplied by 4 + 4. Simplifying the result, we get 2 divided by. 4 multiplied by 8, which is 32. And now simplifying, this is 1 divided by 16. Therefore, the correct answer to this problem corresponds to the answer choice B. 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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Problem 2.4.31b

Textbook Question

Determine the following limits.

b. $\lim_{x \to 3} \frac{x - 3}{x^{4} - 9 x^{2}}$

Verified step by step guidance

Identify the limit expression: $ \lim_{x \to 3} \frac{x-3}{x^4 - 9x^2} $.

Notice that direct substitution of $ x = 3 $ results in an indeterminate form $ \frac{0}{0} $.

Factor the denominator: $ x^4 - 9x^2 = x^2(x^2 - 9) = x^2(x-3)(x+3) $.

Cancel the common factor $ (x-3) $ from the numerator and the denominator.

Re-evaluate the limit with the simplified expression: $ \lim_{x \to 3} \frac{1}{x^2(x+3)} $.

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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 case, we are interested in the limit of a fraction as x approaches 3, which requires evaluating the function's behavior close to that point.

Factoring Polynomials

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to yield the original polynomial. This technique is essential when evaluating limits, especially when direct substitution leads to indeterminate forms like 0/0. In the given limit, factoring the denominator will help simplify the expression and resolve the limit.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to ambiguous results, such as 0/0 or ∞/∞. These forms require further analysis or manipulation to resolve. In this problem, substituting x = 3 directly into the limit results in an indeterminate form, necessitating the use of algebraic techniques like factoring or L'Hôpital's rule to find the actual limit.

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