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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![**Calculus: Infinite Limits**
**Problem: Determine the infinite limit.**
\[ \lim_{{x \to 9}} \frac{x^2}{{(x - 9)^2}} \]
**Solution Options:**
- \( \infty \) (Correct Option, indicated by a blue circle and a checkmark)
- \( -\infty \)
**Help Options:**
- Need Help? [Read It]
**Additional Features:**
- [+] Show My Work (Optional)
**Explanation:**
To solve this limit, we recognize that as \( x \) approaches 9, the denominator \( (x - 9)^2 \) approaches 0, causing the fraction to blow up to infinity. Since both the numerator and the squared denominator are positive as \( x \) approaches 9, the limit tends to positive infinity \( \infty \).
**Educational Note:**
When determining the limit of a rational function where the denominator approaches 0, it's important to consider the signs of both the numerator and denominator to determine whether the limit approaches positive or negative infinity. In this case, since both parts are squared, the result is always positive, leading us to \( \infty \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5ffd7619-5e31-4318-a07b-e1103188d5bd%2F0f20c4dd-817e-4794-826e-7aa7edaaf679%2Fhk1sj9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculus: Infinite Limits**
**Problem: Determine the infinite limit.**
\[ \lim_{{x \to 9}} \frac{x^2}{{(x - 9)^2}} \]
**Solution Options:**
- \( \infty \) (Correct Option, indicated by a blue circle and a checkmark)
- \( -\infty \)
**Help Options:**
- Need Help? [Read It]
**Additional Features:**
- [+] Show My Work (Optional)
**Explanation:**
To solve this limit, we recognize that as \( x \) approaches 9, the denominator \( (x - 9)^2 \) approaches 0, causing the fraction to blow up to infinity. Since both the numerator and the squared denominator are positive as \( x \) approaches 9, the limit tends to positive infinity \( \infty \).
**Educational Note:**
When determining the limit of a rational function where the denominator approaches 0, it's important to consider the signs of both the numerator and denominator to determine whether the limit approaches positive or negative infinity. In this case, since both parts are squared, the result is always positive, leading us to \( \infty \).
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