$\mathbb{R} \subset \mathbb{C}$ Implies Common Limit Values?

Calculus Level 4

If $A$ is the limit in the set of real numbers, and $B$ is the complex limit in the set of complex numbers given by

$\begin{cases} A &= \lim\limits_{x \to + \infty} e^{-x}\\ B &= \lim \limits_{|z|\to + \infty} e^{-z} \end{cases}$

Which of the following must be true?

A

exists, but

B

does not exist. Both

A

and

B

exist as the same value. Both

A

and

B

do not exist.

B

exists, but

A

does not exist. Both

A

and

B

exist, but both as two distinct values.

1 solution

Patrick Corn
Apr 28, 2021

It's clear that $A = 0,$ but $B$ does not exist. For instance, as $z$ goes to $\infty$ along the positive real axis, $e^{-z}$ goes to $0$ ; but as $z$ goes to $\infty$ along the negative real axis, $e^{-z}$ increases without bound.

R ⊂ C \mathbb{R} \subset \mathbb{C} R ⊂ C Implies Common Limit Values?

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$\mathbb{R} \subset \mathbb{C}$ Implies Common Limit Values?