Write a "DELTA - EPSILON" proof (that explicitly uses the definition of limit) to PROVE: lim (2x - y) = - 3 (x, y)-(1,5)
Write a "DELTA - EPSILON" proof (that explicitly uses the definition of limit) to PROVE: lim (2x - y) = - 3 (x, y)-(1,5)
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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Please please For the first image attached please do the calculations similar to the second image attached
![[15] (4) Write a "DELTA - EPSILON" proof (that explicitly uses the definition of limit) to
PROVE: lim (5x - 3y) = 1
(x, y)-(2, 3)
PROOF: WE WILL SHOW THAT FOR EACH E >0
307
0<√(x-2)² + (7-3)² < 8⇒|5z-3y.
⇒>|5z-3y-1|<£
☆: |5x-3y-1|=|5(x-2)-3(7-3)|
≤ 5|x-2|+ 3|y-3|
≤ 5 √(x-2)² + 3√(y - 3)²
≤ 5√(x-2)² + (y-3)² + 3√(x-2) + (7-3)²
< 8 √ √ (2-2)² + (7-3)² (Q-STRATEGY)
LET S = &
Now, FOR ANY E >0, S = & and
0 < √ √(x-2)²+(7-3)² < S →
8 √ √(x - 2)² + (7-3)² < E
→|5z-3y-1|< £ (By A)
Hence, (By DEFINITION) (2 fim (32-33) = 1 0
√(x-2)³+ (y-3)² < &
8](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9755e55f-b204-4ec5-8a10-7bbad16cf793%2F12153e18-1c8a-439e-ac5a-f58bb1252c2d%2Fglscjs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[15] (4) Write a "DELTA - EPSILON" proof (that explicitly uses the definition of limit) to
PROVE: lim (5x - 3y) = 1
(x, y)-(2, 3)
PROOF: WE WILL SHOW THAT FOR EACH E >0
307
0<√(x-2)² + (7-3)² < 8⇒|5z-3y.
⇒>|5z-3y-1|<£
☆: |5x-3y-1|=|5(x-2)-3(7-3)|
≤ 5|x-2|+ 3|y-3|
≤ 5 √(x-2)² + 3√(y - 3)²
≤ 5√(x-2)² + (y-3)² + 3√(x-2) + (7-3)²
< 8 √ √ (2-2)² + (7-3)² (Q-STRATEGY)
LET S = &
Now, FOR ANY E >0, S = & and
0 < √ √(x-2)²+(7-3)² < S →
8 √ √(x - 2)² + (7-3)² < E
→|5z-3y-1|< £ (By A)
Hence, (By DEFINITION) (2 fim (32-33) = 1 0
√(x-2)³+ (y-3)² < &
8
![[15] (4) Write a "DELTA – EPSILON" proof
(that explicitly uses the definition of limit) to
PROVE: lim (2x - y) = -3
(x, y) (1, 5)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9755e55f-b204-4ec5-8a10-7bbad16cf793%2F12153e18-1c8a-439e-ac5a-f58bb1252c2d%2Fnfvpffo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[15] (4) Write a "DELTA – EPSILON" proof
(that explicitly uses the definition of limit) to
PROVE: lim (2x - y) = -3
(x, y) (1, 5)
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