If z is complex variable, how to prove limz→0z4|z|2=0I have simplified to limz→0z4zz―=limz→0z3z―because |z|2=zz―But how to progress?

Let r,θ∈R and writez=reiθThen, z→0 iff r→0Thus, we can substitute z=reiθ into the simplification that you obtained to getlimr→0(reiθ)3re−iθ=limr→0r2e4iθ=0I prefer this method to the modulus method because it is applicable to many other similar cases.

ΗΙΝΤ|zn|=|z|n and |z4|z2||=|z|2Note that limz→0f(z)=0 if and only if limz→0|f(z)|=0

Let r,θ∈R and writez=reiθThen, z→0 iff r→0Thus, we can substitute z=reiθ into the simplification that you obtained to getlimr→0(reiθ)3re−iθ=limr→0r2e4iθ=0I prefer this method to the modulus method because it is applicable to many other similar cases.

ΗΙΝΤ|zn|=|z|n and |z4|z2||=|z|2Note that limz→0f(z)=0 if and only if limz→0|f(z)|=0

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Perla Galloway

Answered question

2022-04-22

If z is complex variable, how to prove

lim_{z \to 0} \frac{z^{4}}{{| z |}^{2}} = 0

I have simplified to

lim_{z \to 0} \frac{z^{4}}{z} \overset{―}{z} = lim_{z \to 0} \frac{z^{3}}{\overset{―}{z}}

because

{| z |}^{2} = z \overset{―}{z}

But how to progress?

Answer & Explanation

Simone Ali

Beginner2022-04-23Added 18 answers

Let

r, θ \in R

and write

z = r e^{i θ}

Then,

z \to 0

iff

r \to 0

Thus, we can substitute

z = r e^{i θ}

into the simplification that you obtained to get

lim_{r \to 0} \frac{{(r e^{i θ})}^{3}}{r} e^{- i θ} = lim_{r \to 0} r^{2} e^{4 i θ} = 0

I prefer this method to the modulus method because it is applicable to many other similar cases.

despescarwh9

Beginner2022-04-24Added 15 answers

ΗΙΝΤ

| z^{n} | = {| z |}^{n}

and

| \frac{z^{4}}{| z^{2} |} | = {| z |}^{2}

Note that

lim_{z \to 0} f (z) = 0

if and only if

lim_{z \to 0} | f (z) | = 0

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