How do I solve this problem: limz→0Re(z2)z+Im(z)?

Bobbie Comstock

Bobbie Comstock

Answered question


How do I solve this problem: limz0Re(z2)z+Im(z)?

Answer & Explanation

Toni Scott

Toni Scott

Beginner2022-01-17Added 32 answers

Writing z=x+iy for some real x,y, we can rephrase this limit as
This limit exists and equals zero. This follows once we show that
To this end, we use polar coordinates:
Clearly |cos(2θ)| is bounded above by 1. Next, we need to show that (cosθ+sinθ)2+cos2θ attains a positive minimum value; this assures that the trigonometric factor in the limit above is bounded (no division by zero).
To this end, observe that
=32+125sin(2θ+α), for some α
Hence, for any r0, there exists a constant C>0 such that

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