Question

Evaluate the following limit.
lim _(x, y) →(0,0)x/x+3 y

Answer 1

The question seeks the limit of x/(x + 3y) as (x, y) approaches (0,0), which results in an indeterminate form of 0/0, requiring further analysis or use of l'Hôpital's rule for a proper evaluation.

Step-by-step explanation:

The question asks us to evaluate the limit of a function as the variables approach a point. The function given is x/(x + 3y) and we are interested in its behavior as (x, y) approaches (0,0).

To evaluate this limit, we would usually check the limit from different paths to see if the function approaches the same value.

However, in this case, it is clear that if we substitute x = 0 and y = 0 directly into the function, we will get 0/0, which is an indeterminate form.

To properly evaluate the limit, we would need additional information or methods like l'Hôpital's rule, which is usually taught in calculus courses.

Answer 2

The evaluate the limit lim _(x, y) →(0,0)x/x+3y, the limit along the x-axis is 1, while the limit along the y-axis is 0. Therefore, the overall limit does not exist.

Step-by-step explanation:

To evaluate the limit lim _(x, y) →(0,0)x/x+3y, we need to consider the behavior of the function as (x, y) approaches (0, 0).

We can approach the limit by taking different paths, such as along the x-axis, y-axis, or any other path in the xy-plane.

Let's consider the limit along the x-axis first.

If we approach (0, 0) along the x-axis, then y=0.

Substituting this into the limit expression, we get x/(x + 3*0) = x/x = 1. Therefore, the limit along the x-axis is 1.

Similarly, if we approach (0, 0) along the y-axis, then x=0. Substituting this into the limit expression, we get 0/(0 + 3y) = 0/y = 0.

Therefore, the limit along the y-axis is 0.

Since the limit values differ depending on the path taken, the overall limit does not exist. So therefore we can conclude that the limit lim _(x, y) →(0,0)x/x+3y does not exist.