Given that lim x→1 (5x − 3) = 2, illustrate Definition 2 by finding values of δ that correspond to ε = 0.1, ε = 0.05, and ε = 0.01.

Question

ε = 0.1 δ≤ =

ε = 0.05 δ≤ =

ε = 0.01 δ≤ =

Answer 1

Given ε = 0.1, find δ > 0 so that if 0 < l x - 1 l < δ, then l (5x-3) - 2 l < 0.1.

Let δ = 0.1 / 5 = 0.02.

If 0 < l x - 1 l < 0.02, then l (5x - 3) - 2 l = l 5x - 5 l = 5 l x - 1 l < 5(0.02) = 0 .1

For each of the other parts, δ = ε/5 works in a similar manner. Note that 5 is the coefficient of x in the definition of f(x). In general, for f(x) = ax + b where a > 0, ε/a is the δ that works in the limit definition.