Answer to Question #160022 in Real Analysis for Luna

Question

In applying the ε-δ definition of continuity to the function f(x)=8x-5 at x=1, given ε>0, we can choose

a) δ=8/ε

b) δ=ε/8

c) δ=5/ε

d) δ=ε/5

e) δ=8ε

Answer 1

Let $\varepsilon>0$ is given. Choose $\delta=\varepsilon/8.$

Assume $|x-1|<\delta.$

Thus

$$|(8x-5)-3|=|8x-8|$$

$$=|8||x-1$$

$$=8|x-1|$$

$$<8\delta$$

$$=8\cdot\dfrac{\varepsilon}{8}$$

$$=\varepsilon$$

Therefore

$$\lim\limits_{x\to1}(8x-5)=3=8(1)-5=f(1)$$

b) δ=ε/8