Final answer:
The limit \( \lim_{(x,y) \to (0,0)} \frac{2x^2 + 3xy + 4y^2}{3x^2 + 5y^2} \) does not exist because the function approaches different values along different paths towards (0,0).
Step-by-step explanation:
The student has asked whether the limit \( \lim_{(x,y) \to (0,0)} \frac{2x^2 + 3xy + 4y^2}{3x^2 + 5y^2} \) exists. To determine the existence of a limit for a function of two variables as both variables approach zero, we must check if the function approaches the same value irrespective of the path taken towards (0,0).
To investigate the limit, we can approach (0,0) along different paths. For example, as a first step, we might evaluate the limit as x approaches 0, which simplifies the function to \( \lim_{y \to 0} \frac{4y^2}{5y^2} = \lim_{y \to 0} \frac{4}{5} \), which equals 4/5. Similarly, if we approach (0,0) along the line y = x, the function simplifies to \( \lim_{x \to 0} \frac{2x^2 + 3x^2 + 4x^2}{3x^2 + 5x^2} = \lim_{x \to 0} \frac{9}{8} \), which equals 9/8. The differing results for these two paths suggest that the limit does not exist as the function approaches different values along different paths.
Therefore, the correct answer to the student's question is No, the limit does not exist.