lim (x,y)->(0,0) of (y-sin(y))/(x^2+y^2)

Question

I have this function

for (x,y)=(0,0) f=0

and on the other f=(y-sin(y))/(x^2+y^2)

is lim (x,y)->(0,0) of (y-sin(y))/(x^2+y^2) = 0 ?

according to wolfarm no but can someone show me why?

Answer 1

let y = x.

 

 The function becomes (x - sinx)/(2*x^2). 

As X tends to zero, so does y.  The limit in 

this case is 0/0 so we can use L'Hopital's rule.

Taking the derivative of top and bottom:

 

      (1 - cos x)/(4x) as x tends to zero.

 

It is still indeterminant 0/0, so we apply

L'Hopitals rule again:

 

          sin x / 4

which tends to zero as x tends to zero.

 

Now let y = e^x - 1 which also tends to zero and x tends to 0

The function becomes:

 

    (e^x - 1 - sin(e^x-1)) /  (x^2 + (e^x-1)^2)

 

Applying L'Hopital's rule

 

   (e^x - cos(e^x-1)*e^x) / (2x + 2(e^x-1)*e^x)

 

   ( 1 - cos(0)*1 ) / (0 + 2*(1-1)*1)

 

  which is still indeterminant 0/0

Applying L'Hopital's rule again:

 

(e^x + sin(e^x-1)*e^x) / (2 +  2(e^x-1)*e^x + e^x + 2e^x)

 

1/(2 + 1 + 2) = 1/5

 

Therefore, by the TWO PATH RULE,

the limit does not exist because there

exists two functions y=x and y=e^x-1

which tend to zero as x tends to zero

but produce different limits.

 

This is the same concept as

in the 2-D case where the

left and right hand limits must

agree in order for the limit to exist