In the epsilon-delta definition, which statement is true about delta?

In the epsilon-delta definition, which statement is true about delta?

• A. delta is chosen before epsilon.
• B. delta is chosen independently of epsilon.
• C. delta is chosen as a function of epsilon.
• D. delta is chosen after epsilon is given.

Answer: (D)

In the epsilon-delta definition, you first fix an epsilon > 0 and then determine a delta > 0 that works for that specific epsilon. This ordering is what lets delta depend on epsilon to ensure that whenever 0 < |x - a| < delta, we have |f(x) - f(a)| < epsilon. So delta is chosen after epsilon is given. While delta can be viewed as a function of epsilon, the crucial point is that the choice is made in response to the fixed epsilon to guarantee the bound.

In the epsilon-delta definition, you first fix an epsilon > 0 and then determine a delta > 0 that works for that specific epsilon. This ordering is what lets delta depend on epsilon to ensure that whenever 0 < |x - a| < delta, we have |f(x) - f(a)| < epsilon. So delta is chosen after epsilon is given. While delta can be viewed as a function of epsilon, the crucial point is that the choice is made in response to the fixed epsilon to guarantee the bound.