- #1
Saladsamurai
- 3,020
- 7
Homework Statement
Given f(x) = mx + b, m > 0, L = (m/2) + b, xo = 1/2 , [itex]\epsilon = c >0[/itex] find (a) an open interval on which the inequality
|f(x) - L| < [itex]\epsilon[/itex]
holds. Then find (b) [itex]\delta[/itex] such that 0 < |x - xo| < [itex]\delta\Rightarrow[/itex] |f(x) - L| < [itex]\epsilon[/itex]
Here is my problem with the book's solution. Since the condition [itex]\epsilon=c>0[/itex] was given, I only used the right-hand-side of the inequality:
[itex]-c<|f(x)-L|<c[/itex] because to me it did not make sense to solve the inequality under a condition that cannot be. Instead, I chose to write the above inequality as:
[itex]0<|f(x)-L|<c[/itex]
But the text gave answer of (a) [itex](\frac{1}{2}-\frac{c}{m}, \frac{c}{m}+\frac{1}{2})[/itex] and (b) [itex]\delta = c/m[/itex]
Why did they use the left-hand-side of the inequality if it was given that c > 0 ?