Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Problem Statement
Calculate the value of the following limit:
\[
\lim_{x \rightarrow 2} \frac{4x^2 - 16}{x - 2}
\]
### Explanation
To find this limit, we can take the following steps:
1. **Factor the Numerator**: Notice that \(4x^2 - 16\) is a difference of squares, which can be written as:
\[
4x^2 - 16 = (2x)^2 - (4)^2 = (2x - 4)(2x + 4)
\]
Therefore, the expression becomes:
\[
\frac{(2x - 4)(2x + 4)}{x - 2}
\]
2. **Simplify the Expression**: We can factor out the expression further:
\[
(2x - 4) = 2(x - 2)
\]
Substitute back:
\[
\frac{2(x - 2)(2x + 4)}{x - 2} = 2(2x + 4)
\]
Note: The \(x - 2\) terms cancel out, simplifying the expression.
3. **Evaluate the Limit**: Now, substitute \(x = 2\) in the simplified expression:
\[
2(2(2) + 4) = 2(4 + 4) = 2 \times 8 = 16
\]
### Conclusion
The value of the limit is \(16\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8c5dd2c-3d76-4b42-b1f3-318d493ae214%2F0ba4106e-e066-4073-bad9-fc85c265388b%2F8wlh6pq_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Calculate the value of the following limit:
\[
\lim_{x \rightarrow 2} \frac{4x^2 - 16}{x - 2}
\]
### Explanation
To find this limit, we can take the following steps:
1. **Factor the Numerator**: Notice that \(4x^2 - 16\) is a difference of squares, which can be written as:
\[
4x^2 - 16 = (2x)^2 - (4)^2 = (2x - 4)(2x + 4)
\]
Therefore, the expression becomes:
\[
\frac{(2x - 4)(2x + 4)}{x - 2}
\]
2. **Simplify the Expression**: We can factor out the expression further:
\[
(2x - 4) = 2(x - 2)
\]
Substitute back:
\[
\frac{2(x - 2)(2x + 4)}{x - 2} = 2(2x + 4)
\]
Note: The \(x - 2\) terms cancel out, simplifying the expression.
3. **Evaluate the Limit**: Now, substitute \(x = 2\) in the simplified expression:
\[
2(2(2) + 4) = 2(4 + 4) = 2 \times 8 = 16
\]
### Conclusion
The value of the limit is \(16\).
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