Problem 30 - Calculus Early Transcendentals

Question

Evaluate the following limits. \(\lim _{x \rightarrow 2}\left(x^{2}-x\right)^{5}\)

Step-by-Step Solution

Answer

Answer: The limit of the function \((x^2-x)^5\) as \(x\) approaches \(2\) is 32.

Step 1: Plug in the given value

We are asked to find the limit as \(x\) approaches \(2\). Let's plug \(x=2\) into the given function: \((2^2 - 2)^5\)

Step 2: Evaluate the expression

Evaluating this expression: \((4 - 2)^5 = 2^5 = 32\)

Step 3: Write the final answer

The limit of the function \((x^2-x)^5\) as \(x\) approaches \(2\) is 32. So, we have: \(\lim _{x \rightarrow 2}\left(x^{2}-x\right)^{5} = 32\)

Key Concepts

Evaluating Limits

When evaluating limits, we are essentially trying to understand how a function behaves as it approaches a particular point. In simple terms, limits help us investigate what happens to the values of a function as the input, or "x-value," gets closer and closer to a certain number.
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**How Limits Work**
When evaluating limits, we can follow a straightforward sequence:

• Identify the point of interest where \(x\) approaches.
• Substitute the value into the function, if it's straightforward to do so, like in the example problem.
• Observe the result to determine the function's behavior as \(x\) nears the point.

Remember, a limit focuses on the value a function approaches, not necessarily the value it takes. If directly plugging in the value yields a clear result without ambiguities like division by zero, we often reach the limit rapidly.

Substitution Method

The substitution method involves directly replacing the variable in a function with the number it approaches. This is a simple, yet powerful technique for finding limits, especially when the function is continuous at the point of interest.
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**Using Substitution Efficiently**

• Check if the substitution leads to any undefined operations (e.g., division by zero).
• If no issues arise, compute the function with the substituted value.
• Simplify the resulting expression if possible and note the limit.

This method is quite effective for polynomial functions, as these are generally continuous everywhere, making substitution both valid and direct.

Polynomial Functions

Polynomial functions consist of terms involving variables raised to any whole number power. They have smooth and continuous graphs, making them easy to work with concerning limits.
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**Characteristics of Polynomial Functions**

• They're continuous across their entire domain, meaning no breaks or holes in their graphs.
• Direct substitution for evaluating limits will usually work without any problems.
• The degree of a polynomial indicates the highest power of the variable; in our example, the inner function \((x^2 - x)\) is polynomial.

In terms of limits, polynomials' continuous nature means we can substitute right into them comfortably to find limits, as shown in our exercise.