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The image you provided contains two limit problems. Let's solve each of them step-by-step.

### Problem 4:
\[
\lim_{(x,y) \to (1,2)} \frac{2xy}{x^2 + y^2}
\]

**Step 1: Plug in the limit point (x, y) = (1, 2)**

Substitute \(x = 1\) and \(y = 2\) into the expression:
\[
\frac{2xy}{x^2 + y^2} = \frac{2(1)(2)}{1^2 + 2^2}
\]

**Step 2: Simplify the expression**

Calculate both the numerator and the denominator:
\[
\text{Numerator} = 2 \times 1 \times 2 = 4
\]
\[
\text{Denominator} = 1^2 + 2^2 = 1 + 4 = 5
\]

Thus, the expression simplifies to:
\[
\frac{4}{5}
\]

**Final answer for Problem 4:**
\[
\boxed{\frac{4}{5}}
\]

---

### Problem 5:
\[
\lim_{(x, y) \to (0, 1)} \left[e^{x^2 + x} \ln(ey^2)\right]
\]

**Step 1: Simplify the expression**

First, recall the property of logarithms: \(\ln(e) = 1\). This simplifies \(\ln(ey^2)\) as:
\[
\ln(ey^2) = \ln(e) + \ln(y^2) = 1 + 2\ln(y)
\]
Thus, the limit becomes:
\[
\lim_{(x, y) \to (0, 1)} e^{x^2 + x} \left(1 + 2\ln(y)\right)
\]

**Step 2: Evaluate the limit**

Now, plug in \(x = 0\) and \(y = 1\):

1. For \(e^{x^2 + x}\), when \(x = 0\), we get:
\[
e^{0^2 + 0} = e^0 = 1
\]

2. For \(1 + 2\ln(y)\), when \(y = 1\), we get:
\[
1 + 2\ln(1) = 1 + 2(0) = 1
\]

Thus, the entire expression becomes:
\[
1 \times 1 = 1
\]

**Final answer for Problem 5:**
\[
\boxed{1}
\]

---

Do you have any questions or would you like a more detailed explanation of any step?

### Related Questions:
1. How do you handle limits approaching infinity for multi-variable functions?
2. Can we use polar coordinates to solve limits for certain multi-variable functions?
3. What are some common indeterminate forms encountered in multi-variable limits?
4. How does L'Hôpital's rule extend to multi-variable calculus?
5. How can you evaluate limits involving trigonometric functions in two variables?

### Tip:
Always check if direct substitution works first when solving limits. It often saves time and effort before considering more advanced techniques like L'Hôpital's rule or using polar coordinates.

This page provides detailed step-by-step solutions for evaluating multivariable limits, including problems involving exponential and logarithmic functions. Suitable for students studying college-level calculus. Learn how to solve limits such as lim (x,y) -> (1,2) of 2xy/(x^2 + y^2) and lim (x,y) -> (0,1) of e^(x^2 + x) ln(ey^2).

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