The image contains two limit problems as $(x, y) \to (0,0)$:

1. $\lim_{(x, y) \to (0,0)} \frac{x^2 + y^2}{x + y}$

2. $\lim_{(x, y) \to (0,0)} x e^{x/y}$

Let's evaluate each limit.

1. $\lim_{(x, y) \to (0,0)} \frac{x^2 + y^2}{x + y}$

To evaluate this limit, let's consider approaching $(x, y) \to (0,0)$ along different paths and check if the result is the same:

• Path $y = x$:

  Substituting $y = x$: $\frac{x^2 + y^2}{x + y} = \frac{x^2 + x^2}{x + x} = \frac{2x^2}{2x} = x \to 0 \text{ as } x \to 0.$

• Path $y = -x$:

  Substituting $y = -x$: $\frac{x^2 + y^2}{x + y} = \frac{x^2 + (-x)^2}{x + (-x)} = \frac{x^2 + x^2}{0} = \text{undefined}.$

Since we get different results along different paths, this limit does not exist.

2. $\lim_{(x, y) \to (0,0)} x e^{x/y}$

To evaluate this limit, let’s analyze the behavior of $e^{x/y}$ as $(x, y) \to (0,0)$.

• Path $y = x$:

  Substituting $y = x$: $x e^{x/y} = x e^{x/x} = x e^1 = x e \to 0 \text{ as } x \to 0.$

• Path $y = x^2$:

  Substituting $y = x^2$: $x e^{x/y} = x e^{x/x^2} = x e^{1/x}.$ As $x \to 0$, $e^{1/x}$ grows extremely large, so this term does not approach a finite limit.

Thus, this limit does not exist as well due to different behavior along different paths.

Would you like more details on these calculations or have any questions?

Further Related Questions

1. How do path-dependent limits work for multivariable functions?
2. What methods can we use to evaluate limits that seem indeterminate?
3. Can polar coordinates simplify multivariable limit problems?
4. When is the Squeeze Theorem useful in evaluating multivariable limits?
5. How can we prove the non-existence of a limit rigorously?

Tip:

In multivariable limits, checking along different paths is a common technique to determine if a limit might not exist.