Chapter 1 - Limits and Continuity - 1.4 Limits (Discussed More Rigorously) - Exercises Set 1.4 - Page 87: 2

(a) $(-0.05, 0.05)$ (b) $(-0.005, 0.005)$ (c) $(-0.0006, 0.00006)$

Step 1 We want x such that $∣2𝑥+3−𝑓(0)∣<𝜖⟺∣2𝑥+3−3∣<𝜖$  $⟺∣2𝑥∣<𝜖$ $⟺2∣𝑥∣<𝜖$  $⟺∣𝑥∣<𝜖/2$ $ ⟺𝑥∈(−𝜖/2,𝜖/2)$ STEP 2: Thus, the largest interval centered at $x = 0$ such that $|2x+3− ƒ (0)| <𝜖$ for different 𝜖 is (a) $𝜖= 0.1; (-0.05, 0.05)$ (b) $ 𝜖 =0.01; (-0.005, 0.005)$ (c) $ 𝜖 =0.0012; (-0.0006, 0.00006)$