Final Answer:
The limit lim (x → 0) x²y²ln(x² + y²) does not exist.
Step-by-step explanation:
To find the limit lim (x → 0) x²y²ln(x² + y²) using polar coordinates, we express x and y in terms of polar coordinates. Let x = r cos(θ) and y = r sin(θ), where r is the distance from the origin and θ is the angle with the positive x-axis.
Substituting these expressions into the limit, we get lim (r → 0) (r cos(θ))² (r sin(θ))² ln((r cos(θ))² + (r sin(θ))²). Simplifying further, we have lim (r → 0) r⁴ ln(r²).
As r approaches 0, r⁴ goes to 0 faster than ln(r²) approaches negative infinity. The product of these two functions leads to an indeterminate form (0 × (-∞)). The limit does not exist because the rate at which r⁴ approaches 0 overwhelms the logarithmic term, resulting in an undefined behavior. Therefore, the limit lim (x → 0) x²y²ln(x² + y²) does not exist.