Suppose that f is continuous at (5, 2) and that f(5, y) = y³ for y# 2. What is the value ƒ(5,2)? (Give your answer as a whole or exact number.) f(5,2)=
Suppose that f is continuous at (5, 2) and that f(5, y) = y³ for y# 2. What is the value ƒ(5,2)? (Give your answer as a whole or exact number.) f(5,2)=
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Question:**
Suppose that \( f \) is continuous at \( (5, 2) \) and that \( f(5, y) = y^3 \) for \( y \neq 2 \). What is the value \( f(5, 2) \)?
(Give your answer as a whole or exact number.)
\[ f(5, 2) = \]
**Solution:**
Given that \( f \) is continuous at \( (5, 2) \), we can use the definition of continuity. A function \( f \) is continuous at a point if the limit of \( f \) as it approaches the point from all directions equals the function's value at that point.
In this case, we can find \( \lim_{y \to 2} f(5, y) \).
By substituting \( f(5, y) \) as \( y^3 \), we get:
\[
\lim_{y \to 2} f(5, y) = \lim_{y \to 2} y^3
\]
To solve this limit:
\[
\lim_{y \to 2} y^3 = 2^3 = 8
\]
Since \( f \) is continuous at \( (5, 2) \), the limit as \( y \) approaches 2 must equal the value at \( (5, 2) \):
\[
f(5, 2) = 8
\]
Therefore,
\[
f(5, 2) = 8
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ed4e6f2-ec2f-4b59-835f-0c83565bb723%2Fef2090fb-7119-4e91-87c4-9e88a1167494%2Fvy3zfva_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
Suppose that \( f \) is continuous at \( (5, 2) \) and that \( f(5, y) = y^3 \) for \( y \neq 2 \). What is the value \( f(5, 2) \)?
(Give your answer as a whole or exact number.)
\[ f(5, 2) = \]
**Solution:**
Given that \( f \) is continuous at \( (5, 2) \), we can use the definition of continuity. A function \( f \) is continuous at a point if the limit of \( f \) as it approaches the point from all directions equals the function's value at that point.
In this case, we can find \( \lim_{y \to 2} f(5, y) \).
By substituting \( f(5, y) \) as \( y^3 \), we get:
\[
\lim_{y \to 2} f(5, y) = \lim_{y \to 2} y^3
\]
To solve this limit:
\[
\lim_{y \to 2} y^3 = 2^3 = 8
\]
Since \( f \) is continuous at \( (5, 2) \), the limit as \( y \) approaches 2 must equal the value at \( (5, 2) \):
\[
f(5, 2) = 8
\]
Therefore,
\[
f(5, 2) = 8
\]
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