Mixing
Alternative way to solve a rational limit in $Bbb C$ when factorization and denominator conjugation fail
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Problem
Show that $$lim_zto wF(z) = 6+8i$$ where $F(z)=dfracz^2+7-24iz-3-4i$ and $w=3+4i$ and $z=x+iy$.
Failed Approaches
I called $n=z^2+7-24i$ and $d=z-3-4i$.
Try factoring $n$ into a difference of squares to cancel with $d$; however $sqrt7-24i=4-3i$.
Try evaluating $F(w)$ by direct substitution, which gives $d=0$ and is undefined.
Try evaluating $F(w)$ by considering $dfracnoverline ddoverline d$; however $overline d = (x-3)-i(y-4)$ evaluates to $0$ at $z=3+4i$ which would imply that $F(w)=0.$
Question
The fact that the question stem suggested to parametrize $z$ into real and imaginary parts suggests to me that I should invoke some theorem about $f(z)to L$ if $renewcommandReoperatornameRe Re f(z) to Re L$ and $renewcommandImoperatornameIm Im f(z) to Im L$.
What other approaches are there to a problem like this? What step might be next with this particular limit?
complex-analysis limits
asked Jul 15 at 16:05
Chase Ryan Taylor
4,24221530
add a comment |Â
up vote
1
down vote
favorite
Problem
Show that $$lim_zto wF(z) = 6+8i$$ where $F(z)=dfracz^2+7-24iz-3-4i$ and $w=3+4i$ and $z=x+iy$.
Failed Approaches
I called $n=z^2+7-24i$ and $d=z-3-4i$.
Try factoring $n$ into a difference of squares to cancel with $d$; however $sqrt7-24i=4-3i$.
Try evaluating $F(w)$ by direct substitution, which gives $d=0$ and is undefined.
Try evaluating $F(w)$ by considering $dfracnoverline ddoverline d$; however $overline d = (x-3)-i(y-4)$ evaluates to $0$ at $z=3+4i$ which would imply that $F(w)=0.$
Question
The fact that the question stem suggested to parametrize $z$ into real and imaginary parts suggests to me that I should invoke some theorem about $f(z)to L$ if $renewcommandReoperatornameRe Re f(z) to Re L$ and $renewcommandImoperatornameIm Im f(z) to Im L$.
What other approaches are there to a problem like this? What step might be next with this particular limit?
complex-analysis limits
asked Jul 15 at 16:05
Chase Ryan Taylor
4,24221530
1
You can just write $F$ as a polynomial. Most straightforward. Your first approach will work, but you have to do it correctly
– Rumpelstiltskin
Jul 15 at 16:08
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Problem
Show that $$lim_zto wF(z) = 6+8i$$ where $F(z)=dfracz^2+7-24iz-3-4i$ and $w=3+4i$ and $z=x+iy$.
Failed Approaches
I called $n=z^2+7-24i$ and $d=z-3-4i$.
Try factoring $n$ into a difference of squares to cancel with $d$; however $sqrt7-24i=4-3i$.
Try evaluating $F(w)$ by direct substitution, which gives $d=0$ and is undefined.
Try evaluating $F(w)$ by considering $dfracnoverline ddoverline d$; however $overline d = (x-3)-i(y-4)$ evaluates to $0$ at $z=3+4i$ which would imply that $F(w)=0.$
Question
The fact that the question stem suggested to parametrize $z$ into real and imaginary parts suggests to me that I should invoke some theorem about $f(z)to L$ if $renewcommandReoperatornameRe Re f(z) to Re L$ and $renewcommandImoperatornameIm Im f(z) to Im L$.
What other approaches are there to a problem like this? What step might be next with this particular limit?
complex-analysis limits
asked Jul 15 at 16:05
Chase Ryan Taylor
4,24221530
Problem
Show that $$lim_zto wF(z) = 6+8i$$ where $F(z)=dfracz^2+7-24iz-3-4i$ and $w=3+4i$ and $z=x+iy$.
Failed Approaches
I called $n=z^2+7-24i$ and $d=z-3-4i$.
Try factoring $n$ into a difference of squares to cancel with $d$; however $sqrt7-24i=4-3i$.
Try evaluating $F(w)$ by direct substitution, which gives $d=0$ and is undefined.
Try evaluating $F(w)$ by considering $dfracnoverline ddoverline d$; however $overline d = (x-3)-i(y-4)$ evaluates to $0$ at $z=3+4i$ which would imply that $F(w)=0.$
Question
The fact that the question stem suggested to parametrize $z$ into real and imaginary parts suggests to me that I should invoke some theorem about $f(z)to L$ if $renewcommandReoperatornameRe Re f(z) to Re L$ and $renewcommandImoperatornameIm Im f(z) to Im L$.
What other approaches are there to a problem like this? What step might be next with this particular limit?
complex-analysis limits
asked Jul 15 at 16:05
Chase Ryan Taylor
4,24221530
asked Jul 15 at 16:05
Chase Ryan Taylor
4,24221530
asked Jul 15 at 16:05
Chase Ryan Taylor
4,24221530
asked Jul 15 at 16:05
Chase Ryan Taylor
4,24221530
4,24221530
1
You can just write $F$ as a polynomial. Most straightforward. Your first approach will work, but you have to do it correctly
– Rumpelstiltskin
Jul 15 at 16:08
add a comment |Â
1
You can just write $F$ as a polynomial. Most straightforward. Your first approach will work, but you have to do it correctly
– Rumpelstiltskin
Jul 15 at 16:08
1
1
You can just write $F$ as a polynomial. Most straightforward. Your first approach will work, but you have to do it correctly
– Rumpelstiltskin
Jul 15 at 16:08
You can just write $F$ as a polynomial. Most straightforward. Your first approach will work, but you have to do it correctly
– Rumpelstiltskin
Jul 15 at 16:08
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
1
down vote
accepted
Hint: Use L'Hospital's rule, then
$$lim_zto3+4idfracz^2+7-24iz-3-4i=lim_zto3+4i2z=6+8i$$
edited Jul 15 at 16:42
Chase Ryan Taylor
4,24221530
answered Jul 15 at 16:17
Nosrati
19.9k41644
Tanks for edit!
– Nosrati
Jul 15 at 16:45
add a comment |Â
up vote
1
down vote
$$F(z) = fracz^2+7-24iz-3-4i = frac(z-3-4i)(z+3+4i)z-3-4i = z+3+4i $$
So that
$$lim_zto 3+4i F(z) = lim_zto 3+4i (z+3+4i) = 6+8i $$
answered Jul 15 at 16:15
Rumpelstiltskin
1,524315
add a comment |Â
up vote
1
down vote
A problem that gives you the answer! Instead of telling you to find
$$lim_zto3+4ifracz^2+7-24iz-(3+4i)$$
it even tells you what that answer should be! If the answer is going to be
$6+8i$ then the numerator had better be
$$(z-(3+4i))(z-(3+4i)+6+8i).$$
Is it? Does that equal $z^2+7-24i$?
answered Jul 15 at 16:15
Lord Shark the Unknown
85.8k951112
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Hint: Use L'Hospital's rule, then
$$lim_zto3+4idfracz^2+7-24iz-3-4i=lim_zto3+4i2z=6+8i$$
edited Jul 15 at 16:42
Chase Ryan Taylor
4,24221530
answered Jul 15 at 16:17
Nosrati
19.9k41644
Tanks for edit!
– Nosrati
Jul 15 at 16:45
add a comment |Â
up vote
1
down vote
accepted
Hint: Use L'Hospital's rule, then
$$lim_zto3+4idfracz^2+7-24iz-3-4i=lim_zto3+4i2z=6+8i$$
edited Jul 15 at 16:42
Chase Ryan Taylor
4,24221530
answered Jul 15 at 16:17
Nosrati
19.9k41644
Tanks for edit!
– Nosrati
Jul 15 at 16:45
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hint: Use L'Hospital's rule, then
$$lim_zto3+4idfracz^2+7-24iz-3-4i=lim_zto3+4i2z=6+8i$$
edited Jul 15 at 16:42
Chase Ryan Taylor
4,24221530
answered Jul 15 at 16:17
Nosrati
19.9k41644
Hint: Use L'Hospital's rule, then
$$lim_zto3+4idfracz^2+7-24iz-3-4i=lim_zto3+4i2z=6+8i$$
edited Jul 15 at 16:42
Chase Ryan Taylor
4,24221530
answered Jul 15 at 16:17
Nosrati
19.9k41644
edited Jul 15 at 16:42
Chase Ryan Taylor
4,24221530
edited Jul 15 at 16:42
Chase Ryan Taylor
4,24221530
edited Jul 15 at 16:42
Chase Ryan Taylor
4,24221530
4,24221530
answered Jul 15 at 16:17
Nosrati
19.9k41644
answered Jul 15 at 16:17
Nosrati
19.9k41644
answered Jul 15 at 16:17
Nosrati
19.9k41644
19.9k41644
Tanks for edit!
– Nosrati
Jul 15 at 16:45
add a comment |Â
Tanks for edit!
– Nosrati
Jul 15 at 16:45
Tanks for edit!
– Nosrati
Jul 15 at 16:45
Tanks for edit!
– Nosrati
Jul 15 at 16:45
add a comment |Â
up vote
1
down vote
$$F(z) = fracz^2+7-24iz-3-4i = frac(z-3-4i)(z+3+4i)z-3-4i = z+3+4i $$
So that
$$lim_zto 3+4i F(z) = lim_zto 3+4i (z+3+4i) = 6+8i $$
answered Jul 15 at 16:15
Rumpelstiltskin
1,524315
add a comment |Â
up vote
1
down vote
$$F(z) = fracz^2+7-24iz-3-4i = frac(z-3-4i)(z+3+4i)z-3-4i = z+3+4i $$
So that
$$lim_zto 3+4i F(z) = lim_zto 3+4i (z+3+4i) = 6+8i $$
answered Jul 15 at 16:15
Rumpelstiltskin
1,524315
add a comment |Â
up vote
1
down vote
up vote
1
down vote
$$F(z) = fracz^2+7-24iz-3-4i = frac(z-3-4i)(z+3+4i)z-3-4i = z+3+4i $$
So that
$$lim_zto 3+4i F(z) = lim_zto 3+4i (z+3+4i) = 6+8i $$
answered Jul 15 at 16:15
Rumpelstiltskin
1,524315
$$F(z) = fracz^2+7-24iz-3-4i = frac(z-3-4i)(z+3+4i)z-3-4i = z+3+4i $$
So that
$$lim_zto 3+4i F(z) = lim_zto 3+4i (z+3+4i) = 6+8i $$
answered Jul 15 at 16:15
Rumpelstiltskin
1,524315
answered Jul 15 at 16:15
Rumpelstiltskin
1,524315
answered Jul 15 at 16:15
Rumpelstiltskin
1,524315
answered Jul 15 at 16:15
Rumpelstiltskin
1,524315
1,524315
add a comment |Â
add a comment |Â
up vote
1
down vote
A problem that gives you the answer! Instead of telling you to find
$$lim_zto3+4ifracz^2+7-24iz-(3+4i)$$
it even tells you what that answer should be! If the answer is going to be
$6+8i$ then the numerator had better be
$$(z-(3+4i))(z-(3+4i)+6+8i).$$
Is it? Does that equal $z^2+7-24i$?
answered Jul 15 at 16:15
Lord Shark the Unknown
85.8k951112
add a comment |Â
up vote
1
down vote
A problem that gives you the answer! Instead of telling you to find
$$lim_zto3+4ifracz^2+7-24iz-(3+4i)$$
it even tells you what that answer should be! If the answer is going to be
$6+8i$ then the numerator had better be
$$(z-(3+4i))(z-(3+4i)+6+8i).$$
Is it? Does that equal $z^2+7-24i$?
answered Jul 15 at 16:15
Lord Shark the Unknown
85.8k951112
add a comment |Â
up vote
1
down vote
up vote
1
down vote
A problem that gives you the answer! Instead of telling you to find
$$lim_zto3+4ifracz^2+7-24iz-(3+4i)$$
it even tells you what that answer should be! If the answer is going to be
$6+8i$ then the numerator had better be
$$(z-(3+4i))(z-(3+4i)+6+8i).$$
Is it? Does that equal $z^2+7-24i$?
answered Jul 15 at 16:15
Lord Shark the Unknown
85.8k951112
A problem that gives you the answer! Instead of telling you to find
$$lim_zto3+4ifracz^2+7-24iz-(3+4i)$$
it even tells you what that answer should be! If the answer is going to be
$6+8i$ then the numerator had better be
$$(z-(3+4i))(z-(3+4i)+6+8i).$$
Is it? Does that equal $z^2+7-24i$?
answered Jul 15 at 16:15
Lord Shark the Unknown
85.8k951112
answered Jul 15 at 16:15
Lord Shark the Unknown
85.8k951112
answered Jul 15 at 16:15
Lord Shark the Unknown
85.8k951112
answered Jul 15 at 16:15
Lord Shark the Unknown
85.8k951112
85.8k951112
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2852645%2falternative-way-to-solve-a-rational-limit-in-bbb-c-when-factorization-and-den%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
You can just write $F$ as a polynomial. Most straightforward. Your first approach will work, but you have to do it correctly
– Rumpelstiltskin
Jul 15 at 16:08