Please help me solve a multivariable limit question

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Homework Help Overview

The discussion revolves around finding the limit of a multivariable function as (r, θ) approaches (0, π/2). The function involves constants w and Vs/Vr, and participants are exploring the implications of these variables in the limit process.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss using L'Hôpital's rule and multiplying by cos(θ) to simplify the expression. There are questions about how to handle the powers of (secθ + tanθ) and the implications of approaching π/2.

Discussion Status

Some participants have provided suggestions for manipulation of the function, while others express confusion about the number of unknowns and how to proceed with the limit. There is an ongoing exploration of the behavior of the function as θ approaches π/2, with no clear consensus on the next steps.

Contextual Notes

Participants note the challenge of determining the value of Vs/Vr and its impact on the limit, as well as the lack of specific values for r and w. The discussion reflects the constraints of the problem setup and the need for further clarification on the constants involved.

CR9
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Homework Statement



Find limit as (r, θ)----> (0, pi/2) for the function:

r= (w secθ)/(secθ+tanθ)^(Vs/Vr)

Both w and Vs/Vr are constants in this question


The Attempt at a Solution



I tried with L'hopital but it didnt turn well as when I differentiate secθ, I got ln (secθ + tan θ)

which when θ approaches 0 still gives me infinite..

Please help me...

I've tried so long to get the answer, but nothing seems to work and I have to pass this up tommorow.
 
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Hi CR9! :smile:

(have a pi: π :wink:)
CR9 said:
Find limit as (r, θ)----> (0, pi/2) for the function:

r= (w secθ)/(secθ+tanθ)^(Vs/Vr)

Both w and Vs/Vr are constants in this question

(you mean "as θ----> π/2" ?)

Try multiplying top and bottom by cosθ :wink:
 
Hi tiny Tim,

Thanks for the reply, yea I mean ( "as θ----> π/2" ?)

Multiplying cos top and bottom would cancel off the sec on top, but how do i times cos inside the denominator? It has power of Vs/Vr

Please advice.

Thanks
 
Hi CR9! :smile:

(secθ+tanθ)Vs/Vr = (secθ+tanθ)(secθ+tanθ)Vs/Vr - 1 :wink:
 
Hi Tiny Tim,

Thanks again for quick reply. You rock!

okay, so after multiplying cos top and bottom and expanding sec+ tan at the bottom as your previous post;

I got:

r/w=1/(1+sinθ )((secθ+tanθ)^(Vs/Vr-1) )
 
But there are too many unknowns, how can I solve this in order to find a value for Vs/Vr.

I need to find the value for Vs/Vr and then find the angle.

Please advice, tim
 
Hi CR9! :smile:
CR9 said:
okay, so after multiplying cos top and bottom and expanding sec+ tan at the bottom as your previous post;

I got:

r/w=1/(1+sinθ )((secθ+tanθ)^(Vs/Vr-1) )

or even 1/(1+sinθ)Vs/Vr(secθ)Vs/Vr - 1
CR9 said:
But there are too many unknowns, how can I solve this in order to find a value for Vs/Vr.

I need to find the value for Vs/Vr and then find the angle.

no problemo … Vs/Vr is a constant, and θ -> π/2

(so, for example, the (1+sinθ) at the beginning obviously –> 2)

Deal with the three cases separately: Vs/Vr > = or < 1 :wink:
 
Hi Tiny Tim,
I was waiting for you online just now...

Anyway, I am still stuck at r/w=1/(1+sinθ )((secθ+tanθ)^(Vs/Vr-1)

What do i do from here?
 
CR9 said:
Hi Tiny Tim,
I was waiting for you online just now...

Anyway, I am still stuck at r/w=1/(1+sinθ )((secθ+tanθ)^(Vs/Vr-1)

What do i do from here?

Hi CR9! :smile:

I don't understand why you're stuck. :confused:

What does (1+sinθ ) tend to as θ -> π/2 ?

And what does (secθ+tanθ) tend to?
 
  • #10
as θ -> π/2,
(1+sin θ) approaches 2
(sec θ + tan θ) approaches infinite?

then from
r/w=1/(1+sinθ )((secθ+tanθ)^(Vs/Vr-1)

it will be
r/w= 1/(2)(infinite)^(vs/vr)-1

What do I do with this?

Please help :(
 
  • #11
Well, what's ∞Vs/Vr-1 ?
 
  • #12
I don't know, Vs/vr could = 0 and it will become 1...

I don't have values for Vs/Vr, r and w...

Please help
 

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