Welcome to EDAboard.com

Welcome to our site! EDAboard.com is an international Electronics Discussion Forum focused on EDA software, circuits, schematics, books, theory, papers, asic, pld, 8051, DSP, Network, RF, Analog Design, PCB, Service Manuals... and a whole lot more! To participate you need to register. Registration is free. Click here to register now.

Help me solve this function limit puzzle

Status
Not open for further replies.

jraks

Newbie level 6
Joined
Dec 28, 2005
Messages
14
Helped
0
Reputation
0
Reaction score
0
Trophy points
1,281
Activity points
1,417
hi guys got a little query

is limit of a function consisting of two variables defined when both the variables are tending to a value simultaneously
eg: lim (x/y)
x->infinity
y->0
just give it atry and if this is wrong just justify
thanks
P.S tis ones for you neils_arm_strong
 

elmolla

Full Member level 2
Joined
Jul 14, 2005
Messages
132
Helped
20
Reputation
40
Reaction score
2
Trophy points
1,298
Activity points
2,721
Re: limit puzzle

Simple :)
 

jraks

Newbie level 6
Joined
Dec 28, 2005
Messages
14
Helped
0
Reputation
0
Reaction score
0
Trophy points
1,281
Activity points
1,417
limit puzzle

isnt infinity/0 undefined
think anout it
 

goumaiy

Newbie level 3
Joined
Feb 17, 2006
Messages
4
Helped
0
Reputation
0
Reaction score
0
Trophy points
1,281
Activity points
1,313
Re: limit puzzle

Yes, lim (x/y)=∞. If this is not clear, just switch the denominator and the numerator. You get lim (y/x)=0. Thus, lim (x/y)=∞.

This (∞/0, 0/∞) is perfectly defined. ∞/∞ and 0/0 are undefined.
 

mobile

Newbie level 1
Joined
Feb 2, 2006
Messages
1
Helped
0
Reputation
0
Reaction score
0
Trophy points
1,281
Activity points
1,292
limit puzzle

∞ is correct answer?
 

kishangk

Newbie level 4
Joined
Feb 15, 2006
Messages
7
Helped
0
Reputation
0
Reaction score
0
Trophy points
1,281
Activity points
1,324
Re: limit puzzle

very easy...infinity....:D
 

kar2kn

Newbie level 6
Joined
Feb 20, 2006
Messages
12
Helped
1
Reputation
2
Reaction score
0
Trophy points
1,281
Activity points
1,369
Re: limit puzzle

answer is 0.
the rule is according to L'Hospital .
 

jayc

Member level 3
Joined
Feb 14, 2006
Messages
64
Helped
11
Reputation
22
Reaction score
0
Trophy points
1,286
Activity points
2,007
Re: limit puzzle

L'hospital's does not apply here. The rule is used only if the limit goes to
\[\frac{0}{0}\] or \[\frac{\infty}{\infty}\].
 
Last edited by a moderator:

sp002

Newbie level 4
Joined
Feb 12, 2006
Messages
7
Helped
0
Reputation
0
Reaction score
0
Trophy points
1,281
Activity points
1,345
Re: limit puzzle

∞ is correct answer.
 

jraks

Newbie level 6
Joined
Dec 28, 2005
Messages
14
Helped
0
Reputation
0
Reaction score
0
Trophy points
1,281
Activity points
1,417
limit puzzle

ok guys how about

lim x^y
x->infinity
y->0

solve it

and on more thing is limit defined at apoint when the two variables are simulataneously varying and tending to some thing?

think
 

jayc

Member level 3
Joined
Feb 14, 2006
Messages
64
Helped
11
Reputation
22
Reaction score
0
Trophy points
1,286
Activity points
2,007
Re: limit puzzle

To solve this, we first need to express the limit in a different form.

Consider the following:
\[\lim\limit_{x\rightarrow\infty, y\rightarrow 0} x^y = \lim\limit_{x\rightarrow\infty, y\rightarrow 0} e^{\ln(x^y)}\]

Let us first find the limit of the exponent:
\[\lim\limit_{x\rightarrow\infty, y\rightarrow 0} \ln x^y = \lim\limit_{x\rightarrow\infty, y\rightarrow 0} y \ln x\]

This equates to 0*inf so we need to apply L'Hospital's Rule:
\[\lim\limit_{x\rightarrow\infty, y\rightarrow 0} \frac{\ln x}{1/y}\]
\[{} = \lim\limit_{x\rightarrow\infty, y\rightarrow 0} \frac{1/x}{-1/y^2} = \lim\limit_{x\rightarrow\infty, y\rightarrow 0} -\frac{y^2}{x}\]

It is now apparent that this limit goes to 0/inf, which is just 0.

Now we can go back to the original limit of interest by raising the result in the base e. This gives the result
\[e^0 = 1\]
 
Last edited by a moderator:

LouisSheffield

Member level 5
Joined
Feb 19, 2006
Messages
82
Helped
10
Reputation
20
Reaction score
0
Trophy points
1,286
Activity points
2,164
Re: limit puzzle

Let's not forget, too, that it hasn't been specified as to whether y->0 from the left OR from the right.
Also, we can't apply L'Hopital's rule unless we have a functional form that relates x and y. For example, if y=1/x, then x/y=x^2, which is always positive (etc).
 

jayc

Member level 3
Joined
Feb 14, 2006
Messages
64
Helped
11
Reputation
22
Reaction score
0
Trophy points
1,286
Activity points
2,007
Re: limit puzzle

LouisSheffield said:
Let's not forget, too, that it hasn't been specified as to whether y->0 from the left OR from the right.
Also, we can't apply L'Hopital's rule unless we have a functional form that relates x and y. For example, if y=1/x, then x/y=x^2, which is always positive (etc).

In this problem, it doesn't matter from which direction y approaches the limit, since there is no divergence at the limit case. Also, we do not need to know a relation between x and y. This is because the numerator and denominator only contain one term and are completely independent. L'Hospital's Rule is useful because it determines the limit by finding which term (numerator or denominator) changes faster. Since they are independent, there is no problem with taking the derivative of the numerator with respect to x and the denominator with respect to y.
 

LouisSheffield

Member level 5
Joined
Feb 19, 2006
Messages
82
Helped
10
Reputation
20
Reaction score
0
Trophy points
1,286
Activity points
2,164
Re: limit puzzle

But there is divergence - x is clearly positive, but not y.
At least y was not stated as being positive as it approaches 0.

As for L'Hopital, one must have a function in order to differentiate said function.

Also, the function you specified above (x^y) is not what the original question involved. (the example was x/y, which is divergent)


Just noticed - Sorry about any confusion - my post was to the original problem x/y,
not the solution you formulated a few posts ago.
 

Status
Not open for further replies.

Similar threads

Part and Inventory Search

Welcome to EDABoard.com

Sponsor

Top