Question

For every positive number x, there is a positive number Δ such that |x - a| < Δ implies |f(x) - f(a)| < ________.

Answer 1

The epsilon-delta definition of limit states that for every positive number x, there is a positive number Δ such that |x - a| < Δ implies |f(x) - f(a)| < Δ.

Step-by-step explanation:

For every positive number x, there is a positive number Δ such that |x - a| < Δ implies |f(x) - f(a)| < Δ. This statement is known as the epsilon-delta definition of limit.

When we say that |x - a| < Δ, it means that x is within a distance of Δ from a. Similarly, when we say that |f(x) - f(a)| < Δ, it means that f(x) is within a distance of Δ from f(a).

The key concept behind this definition is that we can make f(x) arbitrarily close to f(a) by taking x sufficiently close to a (within a distance of Δ)