Find the limit and prove the limit for

Delilah Novak

Delilah Novak

Answered question

2022-04-13

Find the limit and prove the limit for
lim(x,y,z)(1,2,3)arctan(x+zy)

Answer & Explanation

posciad48o

posciad48o

Beginner2022-04-14Added 15 answers

We have
|arctan(x+zy)|arctan(1)|=|arctan(x+y+zxy+z)
=|arctan((x1)+(y2)+(z+3)xy+z)|
|(x1)+(y2)+(z+3)xy+z|
3(x1)2+(y2)2+(z+3)2|xy+z| (1)
First we choose δ=1 so that if (x1)2+(y2)2+(z+3)2<δ=1, then 0<x<2, 3<y<1, and 4<z<2
Therefore, with δ=1, we see that 1|xy+z|<1. Then, from (1) we have
|arctan(x+zy)arctan(1)|3(x1)2+(y2)2+(z+3)2<ϵ
whenever (x1)2+(y2)2+(z+3)2<min(1.ϵ3)

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?