Question

if f(x ,y)=\(\sqrt {4 - {x^2} - 4{y^2}} \)find \(\)\({f_x}\left( {1,0} \right)\) and \({f_y}\left( {1,0} \right)\)and interpret these numbers as slopes .Illustrate with either hand –dream sketches are computer plots.

Short answer

given f(x ,y)=\(\sqrt {4 - {x^2} - 4{y^2}} \) find \({f_x}\left( {1,0} \right)\) and \({f_y}\left( {1,0} \right)\) and interpret these numbers as slopes .Illustrate with either hand –dream sketches are computer plots.

Step-by-step solution

Step1:- given data

Find f(x ,y) find \({f_x}\left( {1,0} \right)\)and \({f_y}\left( {1,0} \right)\)and interpret these numbers as slopes.

Step2:- let us consider 

\(\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{5{y^4}{{\cos }^2}x}}{{{x^4} + {y^4}}}\)

Now verify that the limit exists or not

Let the function be,

Step3:- consider the following limit :

the function f(x ,y) approaches (0,0) along the x-axis. Then y=0 gives

Thus ,f(x , y)→0 along the line y=0

Step4:- consider the limit \(\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{5{y^4}{{\cos }^2}x}}{{{x^4} + {y^4}}}\)

Find the limit

the function f(x ,y) approaches (0,0) along the y-axis. Then x=0 gives

∴ Thus ,f(x , y)→5 along the line x=0

Since the function f has two different line .

therefore, the limit line dose not exist \(\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{5{y^4}{{\cos }^2}x}}{{{x^4} + {y^4}}}\)doesnot exist.