Toronto Math Forum
Welcome,
Guest
. Please
login
or
register
.
1 Hour
1 Day
1 Week
1 Month
Forever
Login with username, password and session length
News:
Home
Help
Search
Calendar
Login
Register
Toronto Math Forum
»
MAT334-2018F
»
MAT334--Tests
»
Quiz-2
»
Q2 TUT 5101
« previous
next »
Print
Pages: [
1
]
Author
Topic: Q2 TUT 5101 (Read 5077 times)
Victor Ivrii
Administrator
Elder Member
Posts: 2607
Karma: 0
Q2 TUT 5101
«
on:
October 05, 2018, 06:15:04 PM »
Find the limit of each function at the given point $z_0$, or explain
why it does not exist.
\begin{equation*}
f(z)=\frac{z^3-8i}{z+2i},\quad z\ne 2i, \qquad\text{at } z_0=2i.
\end{equation*}
Logged
Yatong Yu
Jr. Member
Posts: 6
Karma: 6
Re: Q2 TUT 5101
«
Reply #1 on:
October 05, 2018, 11:05:13 PM »
f(z) = z³+(2i)³i/z+2i
= (z+2i)(z²-2iz+(2i)²)/z+2i
=z²-2iz-4
lim
z->2i
f(z)=lim
z->2i
z²-2iz-4
=(2i)²-2i(2i)-4
= -4+4-4
=-4
Practically unreadable despite all insane html "mathematics".
$$\begin{aligned}f(z) &= \frac{z^3+(2i)^3i}{z+2i}\\
&= \frac{(z+2i)(z^2-2iz+(2i)^2)}{z+2i}\\
&=z^2-2iz-4\\
\lim_{z\to2i} f(z)&=\lim_{z\to2i} z^2-2iz-4\\
&=(2i)^2-2i(2i)-4\\
&= -4+4-4\\
&=-4
\end{aligned}$$
«
Last Edit: October 06, 2018, 05:51:49 AM by Victor Ivrii
»
Logged
Print
Pages: [
1
]
« previous
next »
Toronto Math Forum
»
MAT334-2018F
»
MAT334--Tests
»
Quiz-2
»
Q2 TUT 5101