How can we verify this limit via \epsilon-\delta method? I have



Answered question


How can we verify this limit via ϵδ method?
I have to calculate the limit

Answer & Explanation

Lakisha Archer

Lakisha Archer

Beginner2022-01-13Added 39 answers

We have to show that
ϵ>0:δ(0,π2]:xR {0}
where f(x)=(1cosx)26sin2x=(1cosx)26(1cos2x)
Let t=cosx. Then, because t1 whenever x0 and |x|<π2,
f(x) strictly decreases w.r.t.t.
Find t that makes f(x)=ϵ. After all, t=2(6ϵ+1)1. Since t=cosx, choosing
proves the claim. I think you can show that δ is an increasing function of ϵ.
Mary Nicholson

Mary Nicholson

Beginner2022-01-14Added 38 answers

You can also multiply and divide by 1+cosx:
If you need to use the definition, you can easily see that
Since there is no indetermination in this limit, you can use Heine's definition in a very straightforward way (instead of Cauchy's). Or, you can go on to obtain

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