ASSIGNMENT 9.

NS 101: Mathematics for Continuous Domains

1. Show that

(i) lim(x,y)→(0,0)

√x2y2+1−1

x2+y2 = 0 (ii) lim(x,y)→(2,1)sin−1(xy−2)tan−1(3xy−6)

= 13.

2. Show that the limit when (x, y) → (0, 0) does not exist.

(i) lim x2y2

x2y2+(x2−y2)2

(ii) limx3+y3

x−y(iii) lim xy3

x2+y6

3. Show that the limit exists at the origin but the repeated limits do not exist, where

f(x, y) =

{

xsin(

1y

)

+ ysin(

1x

)

, when xy 6= 0,

0, when xy = 0.

4. Show that the function f(x, y) =

{

x2y2

x2y2+(x−y)2, when (x, y) 6= (0, 0)

0, when (x, y) = (0, 0).satisfy the

following(i) The repeated limits exist and equal to zero.(ii) Limit of function does not exist at (0, 0)(iii) f(x, y) is not continuous at (0, 0)(iv) the partial derivatives existat (0, 0).

5. Let f(x, y) =

{

(x2 + y2)sin 1x2+y2 , when (x, y) 6= (0, 0)

0, when (x, y) = (0, 0).. Show that the function

f if differentiable at (0, 0) but the partial derivatives are not continuous at (0, 0).

6. Let f(x, y) =

{

xy x2−y2

x2+y2 , when (x, y) 6= (0, 0)

0, when (x, y) = (0, 0).. Prove that

(i) fx(0, y) = −y and fy(x, 0) = x for all x and y.(ii) fxy(0, 0) = −1 and fyx(0, 0) = 1(iii) f is differentiable at (0, 0).

7. Show that the function f , where

f(x, y) =

{

xy√x2+y2

, when (x, y) 6= (0, 0)

0, when (x, y) = (0, 0)..

is continuos, possesses partial derivatives but not differentiable at origin.

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