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Math 375 : Assignment 2

Due Thursday, Feb 21

Problem 1: Consider the complex function

f(x, y) =

x

2

y

x

4 + y

2

+ i

xy2

x

2 + y

2

(a) Compute the limit of f as (x, y) ? (0, 0) along the vertical, imaginary axis:

lim

t ? 0

f(0, t)

(b) Compute the limit as (x, y) ? (0, 0) along any (every) other line, y = mx:

lim

t ? 0

f(t, mt)

(c) Your work above should establish the fact that the limit of f along any

straight line to the origin is the same. Despite this, prove that the (full,

two-dimensional, complex) limit does not exist. Hint: Try taking the limit

along a familiar curved path to the origin.

Problem 2: Consider the linear function: f(z) = (3 + 2i)z + (?4 + i)

(a) Prove, analytically, directly using the definition of limit, that

lim z ? 2?i

f(z) = 4 + 2i

Hint: We talked about a simple way to do this for linear functions. But the

more general ‘re-centering’ idea will also work here; see Problem 3. Try it

both ways!

(b) More concretely, if I give you an error radius = 1 around 4 + 2i ? C, what

tolerance radius ? can you take around 2 ? i ? C that ensures

0 < |z ? (2 ? i)| < ? =? |f(z) ? (4 + 2i)| < 1

Explicitly test your answer for ? by picking a concrete point w in the

corresponding punctured tolerance disk around 2 ? i and confirming that

its image f(w) lands inside the error disk around 4 + 2i of radius = 1.

(Compute f(w) explicitly using the formula!) Draw a picture of the

punctured tolerance disk around 2 ? i, showing your point w, and draw

a second picture of the error disk, showing the image f(w). Finally, also

explain your answer for ? geometrically, using the fact that f is linear.

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## Tutor Answer

Hi, here is your assignment :). Everything has done through step by step solution :)

Math 375 : Assignment 2

Problem 1:

a. Along the vertical, imaginary axis:

Given the function, if is along the x-axis then;

lim

(𝑥,𝑦)→(𝑥,0)

𝑥2𝑦

( 4

)=𝟎

(𝑥,𝑦)→(𝑥,0) 𝑥 + 𝑦 2...

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