- #1
ƒ(x)
- 328
- 0
Given: limit of (sin x)/x as x --> 0 and that ε = .01
Problem: Find the greatest c such that δ between zero and c is good. Give an approximation to three decimal places.
Equations:
0 < |x - a| < δ
0 < |f(x) - L| < εAttempt:
0 < |x - 0| < δ
0 < | sin(x)/x - 1| < ε
0 < | sin(x)/x - 1| < .01
0 < | sin(x)/x| < 1.01
0 < |sin(x)| < 1.01|x|
0 < |sin(x)| < 1.01δ
0 < |sin(x)|/1.01 < δ
Since sin(x) is going to range between -1 and 1, the greatest value for δ is 1/1.01. But, this answer isn't correct. The correct answer is .245, and I don't know how to get that.
Problem: Find the greatest c such that δ between zero and c is good. Give an approximation to three decimal places.
Equations:
0 < |x - a| < δ
0 < |f(x) - L| < εAttempt:
0 < |x - 0| < δ
0 < | sin(x)/x - 1| < ε
0 < | sin(x)/x - 1| < .01
0 < | sin(x)/x| < 1.01
0 < |sin(x)| < 1.01|x|
0 < |sin(x)| < 1.01δ
0 < |sin(x)|/1.01 < δ
Since sin(x) is going to range between -1 and 1, the greatest value for δ is 1/1.01. But, this answer isn't correct. The correct answer is .245, and I don't know how to get that.