I want to find this limit for complex

Izabelle Walsh

Izabelle Walsh

Answered question

2022-04-07

I want to find this limit for complex variable z
limz1(z+1)sin(1z+1)
In the real case I know sin(z) is bounded by -1,1, and the limit is 0. But in the complex case sin(z) is not bounded. How can I find the limit or prove that it doesn't exist?

Answer & Explanation

abangan85s0

abangan85s0

Beginner2022-04-08Added 16 answers

One may first consider the sequence {zn} of complex numbers such that
zn+1=1nπ,n,zn1,
giving
limzn1(zn+1)sin(1zn+1)=limn1nπsin(nπ)=0.
Then one may consider the sequence {zn} of complex numbers such that
zn+1=1nπi,n,zn1,
giving
limzn1(zn+1)sin(1zn+1)=limnenπenπ2nπ=
Then considered limit
limz1(z+1)sin(1z+1)
does not exist.

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