Determine the following limits. lim x → 0 x 3 − 5 x 2 x 2 {\displaystyle\lim_{x\to0}}\frac{x^3-5x^2}{x^2}

Welcome back, everyone. Find the limit. Limit as X approaches 0 of 2 X to the power of 4th plus 3X 2 divided by 5X2d. We're given 4 answer choices. A says 5/3, B. 5 halves C 2/5 and D 35. So let's rewrite the limit limit as x approaches 0 of 2 X to the power of 4th plus 3 x 2 divided by 5 x 2. We're going to begin this problem by direct substitution. In the numerator, if we substitute X equals 0, we're going to get 0 + 0. This gives us 0, and in the denominator 5 multiplied by 0 squared, that's 0 as well. So we have an indeterminate form. Whenever we have an indeterminate form, we want to eliminate a potential whole. So let's try factorization. Let's notice that for the numerator we can factor out X2, which gives us 2x2 + 3 as the remaining factor. And in the denominator we have 5 x squared. We can see that we can now cancel out. The X squared and since we have canceled out a potential whole, let's attempt direct substitution once again in the numerator we are going to get 2 multiplied by 0 squared plus 3 and in the denominator we're going to have 5. Now this is going to be a finite number. 0 + 3 is 3. In the denominator we have 5, and since this is now a finite number, we have successfully evaluated the limit. This is 3/5 which corresponds to the answer choice D. 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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2:05 minutes

Problem 2.4.33

Textbook Question

Determine the following limits.

$\lim_{x \to 0} \frac{x^{3} - 5 x^{2}}{x^{2}}$

Verified step by step guidance

Step 1: Identify the limit expression: $ \lim_{x \to 0} \frac{x^3 - 5x^2}{x^2} $.

Step 2: Simplify the expression by factoring out the common term in the numerator. The numerator $ x^3 - 5x^2 $ can be factored as $ x^2(x - 5) $.

Step 3: Cancel the common factor $ x^2 $ in the numerator and the denominator. This simplifies the expression to $ x - 5 $.

Step 4: Substitute $ x = 0 $ into the simplified expression $ x - 5 $ to evaluate the limit.

Step 5: Conclude the limit by substituting the value of $ x $ into the simplified expression.

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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. For example, the limit of a function as x approaches 0 can reveal the function's value or behavior at that point, even if the function itself is not explicitly defined there.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that can be multiplied together to obtain the original expression. In the context of limits, factoring can simplify complex rational expressions, making it easier to evaluate limits by canceling out common terms. For instance, in the limit problem given, factoring the numerator can help eliminate the indeterminate form that arises when substituting the limit directly.

Indeterminate Forms

Indeterminate forms occur in calculus when evaluating limits leads to expressions that do not provide clear information about the limit's value, such as 0/0 or ∞/∞. These forms require further analysis, often through algebraic manipulation, L'Hôpital's Rule, or other techniques, to resolve. Recognizing an indeterminate form is crucial for applying the appropriate methods to find the actual limit.

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