Please help me solve a multivariable limit question

[CR9]

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.

 

[tiny-tim]

Hi CR9!

(have a pi: π )

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

 

[CR9]

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

 

[tiny-tim]

Hi CR9!

(secθ+tanθ)^Vs/Vr = (secθ+tanθ)(secθ+tanθ)^{Vs/Vr - 1}

 

[CR9]

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) )

 

[CR9]

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

 

[tiny-tim]

Hi CR9!

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}

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 ​

 

[CR9]

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?

 

[tiny-tim]

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!

I don't understand why you're stuck.

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

And what does (secθ+tanθ) tend to?

 

[CR9]

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 :(

 

[tiny-tim]

Well, what's ∞^Vs/Vr-1 ?

 

[CR9]

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