Final answer:
The student's question on verifying a limit appears to have a typo, as it lacks a function f(x, y) and a limit value L. For a correct verification, the definition of a limit must be applied, considering an asymptote and behavior of the potential function near the given point by analyzing (x, y) data pairs.
Step-by-step explanation:
The student has asked to verify the limit lim (x, y) → (1, 4) y = -4 using the definition of the limit of a function of two variables. However, a potential typo may exist because if y approaches -4 as (x, y) approaches (1, 4), it appears inconsistent. Typically, the limit statement would specify a function f(x, y) approaching a particular value L as (x, y) approaches a point. For a proper verification of the limit, we would need a function f(x, y) and need to show that for every ε > 0 there exists a δ > 0 such that for all (x, y), if 0 < √((x - 1)2 + (y - 4)2) < δ, then |f(x, y) - L| < ε. Without the specific function and the value of L, we cannot proceed with the verification.
Considering the concept of limits, it's important to note that the presence of an asymptote in a function, such as y = 1/x where x and y cannot be zero simultaneously, indicates a limit behavior where the function approaches infinity as its variable approaches some value. A function might display similar characteristics when approached along different paths, and this is essential to confirming that a limit exists at a point.
To accurately determine and verify limits, one often plots or sketches the graph of the function using specific (x, y) data pairs to observe the behavior of the function as it approaches the limit point from different directions.