Q 14. - Calculus

Question

Copy the figure that follows onto a sheet of paper. Now cut a slit along the dashed line, leave the left side of the paper on the table, and gently raise the right side of the paper along the slit .

(a) Explain how the paper may be interpreted as the graph of a function of two variables, $f(x,y)$.  

(b) If the tabletop is the $xy$-plane, explain why $\underset{\underset{C_1}{(x,y)\to (a,b)}}{\lim }f(x,y)=\underset{\underset{C_2}{(x,y)\to (a,b)}}{\lim }f(x,y)=0$

(c) Explain why $\underset{\underset{C_3}{(x,y)\to (a,b)}}{\lim }f(x,y)=\underset{\underset{C_4}{(x,y)\to (a,b)}}{\lim }f(x,y)=\underset{\underset{C_5}{(x,y)\to (a,b)}}{\lim }f(x,y)>0$

(d) Explain why $\underset{\underset{}{(x,y)\to (a,b)}}{\lim }f(x,y)$ does not exist.

Step-by-Step Solution

Answer

Part (a): The sheet of paper may be written as $f(x,y)=Ax+By+C$

Part (b):  $C_1 and C_2$ both lie on $xy-plane$

Part (c):  $C_3,C_{4 }and C_5$ are on third axis.

Part (d): At this point the limit is not equal along all curves.

Part (a): Defining the limit

The plane of the sheet of paper is rectangular. A plane is represented by a two-variable linear function. As a result, the sheet of paper may be written as $f(x,y)=Ax+By+C$.

Part (b): Evaluating the limit

If the table top is the $xy$-plane, In this plane, the value of function is always $0$. Both the curves $C_{1 }and C_2$ are in the $xy$-plane. As a result, the function's limiting value along these curves is always $0$. That is $\underset{\underset{C_1}{(x,y)\to (a,b)}}{\lim }f(x,y)=\underset{\underset{C_2}{(x,y)\to (a,b)}}{\lim }f(x,y)=0$

Part (c): Evaluating the limit

The curves $C_3,C_{4 }and C_5$, are are on the third axis, not in the table top plane. As a result, the value of the third variable on the points on these curves would yield the function's value. The $z-values$ would be positive since they are positioned above the table. That is

$\underset{\underset{C_3}{(x,y)\to (a,b)}}{\lim }f(x,y)=\underset{\underset{C_4}{(x,y)\to (a,b)}}{\lim }f(x,y)=\underset{\underset{C_5}{(x,y)\to (a,b)}}{\lim }f(x,y)>0$

Part (d): Evaluating the limit

The limiting value of function along different curves is not equal. As

$\underset{\underset{C_1}{(x,y)\to (a,b)}}{\lim }f(x,y)=\underset{\underset{C_2}{(x,y)\to (a,b)}}{\lim }f(x,y)=0$

And

$\underset{\underset{C_3}{(x,y)\to (a,b)}}{\lim }f(x,y)=\underset{\underset{C_4}{(x,y)\to (a,b)}}{\lim }f(x,y)=\underset{\underset{C_5}{(x,y)\to (a,b)}}{\lim }f(x,y)>0$

Hence the limit of function along this point is not defined.

Thus $\underset{\underset{}{(x,y)\to (a,b)}}{\lim }f(x,y)$ does not exist.