We have to find the value of lim x,y-->0 [(x^3-y^3)/(x^2+y^2)]
We cannot substitute x and y with 0 in the expression as that would yield 0/0 which is indeterminate.
We see that the expression is defined for all values of (x,y) except (0,0), so we approach (0,0) along y = mx
lim x,y-->0 [(x^3-y^3)/(x^2+y^2)]
=> lim x-->0 [(x^3- m^3*x^3)/(x^2+m^2*x^2)]
=> lim x-->0 [(x^3(1 - m^3))/(x^2(1+m^2))]
=> lim x-->0 [(x(1 - m^3))/(1+m^2)]
Now substituting x = 0
=> 0*(1 - m^3) / (1 + m^2)
=> 0
Therefore the required value of the limit is 0.
We’ll help your grades soar
Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.
- 30,000+ book summaries
- 20% study tools discount
- Ad-free content
- PDF downloads
- 300,000+ answers
- 5-star customer support
Already a member? Log in here.
Are you a teacher? Sign up now