Answer:
A. DNE. The limit does not exist.
B. The limit is 0
Step-by-step explanation:
A. lim(x,y)→(0,0)[(x + 5y)/(2x² + 25y²)]
Along the x axis, y = 0, the limit becomes:
lim(x)→(0)[x/2x²]
= lim(x)→(0)[1/2x] = infinity
Along the y axis, x = 0, the limit becomes
lim(y)→(0)[5y/25y²]
= lim(y)→(0)[1/5y] = infinity
DNE
B. lim(x,y)→(0,0)[(2x³ + 7y³)/(x² + y²)
Putting x = rcosθ, and y = rsinθ, the limit becomes
lim(r)→(0)[(2r³cos³θ + 7r³sin³θ)/(r²cos²θ + r²sin²θ)]
Because cos²θ + sin²θ = 1, the limit becomes:
= lim(r)→(0)[r³(2cos³θ + 7sin³θ)/r²]
= lim(r)→(0)[r(2cos³θ + 7sin³θ)]
= 0
The limit is 0
