Question #114915
Using the ε −δ definition, show that lim 3( x )5 1
1
Expert's answer
2020-05-12T17:28:22-0400

We have to prove that given any ε>0ε>0  we can find a δε>0δ_ε>0  such that

f(x)1<εx(2δε,2+δε)| f ( x ) − 1 | < ε \quad \forall x ∈ ( 2 − δ_ ε , 2 + δ_ ε )

Let's evaluate the difference

f(x)1=3x51=3x6=3x2| f ( x ) − 1 | = | 3 x − 5 − 1 | = | 3 x − 6 | = 3 | x − 2 |

Therefore

f(x)1<εx2<ε3| f ( x ) − 1 | < ε ⇔ | x − 2 | < \frac{ε}{3}

Given, that ε>0ε>0 we can choose δε<ε3δ_ε<\frac{ε}{3}

Hence

x(2δε,2+δε)x2<ε3f(x)1<εx ∈ ( 2 − δ_ ε , 2 + δ_ ε ) ⇒ | x − 2 | < \frac{ε}{3} ⇔ | f ( x ) − 1 | < ε



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS