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Let's solve the two given limit problems step by step.

### Problem 1
\[
\lim_{x \to +\infty} x (\sqrt{x^2 + 1 - x})
\]

**Solution:**

First, simplify the expression inside the square root:

\[
\sqrt{x^2 + 1 - x}
\]

As \( x \to +\infty \), the term \( x^2 \) dominates the expression inside the square root. So, we can approximate:

\[
\sqrt{x^2 + 1 - x} \approx \sqrt{x^2 - x} = x\sqrt{1 - \frac{1}{x}} \approx x\left(1 - \frac{1}{2x}\right) = x - \frac{1}{2}
\]

Thus, the expression becomes:

\[
x \left(x - \frac{1}{2}\right) = x^2 - \frac{x}{2}
\]

Taking the limit as \( x \to +\infty \):

\[
\lim_{x \to +\infty} (x^2 - \frac{x}{2}) = \infty
\]

So, the answer for the first limit is:
\[
\infty
\]

### Problem 2
\[
\lim_{x \to +\infty} \left(\frac{2x + 3}{2x + 1}\right)^{x + 1}
\]

**Solution:**

First, analyze the base of the exponent:

\[
\frac{2x + 3}{2x + 1} = \frac{2x (1 + \frac{3}{2x})}{2x (1 + \frac{1}{2x})} = \frac{1 + \frac{3}{2x}}{1 + \frac{1}{2x}}
\]

As \( x \to +\infty \), both \( \frac{3}{2x} \) and \( \frac{1}{2x} \) approach 0:

\[
\frac{1 + \frac{3}{2x}}{1 + \frac{1}{2x}} \to 1
\]

However, this expression converges to 1. To evaluate the limit, consider the expression in the form suitable for applying the exponential limit:

\[
\left(\frac{2x + 3}{2x + 1}\right)^{x + 1}
\]

Let \( y = \left(\frac{2x + 3}{2x + 1}\right)^{x + 1} \). Taking the natural logarithm on both sides:

\[
\ln y = (x + 1) \ln \left(\frac{2x + 3}{2x + 1}\right)
\]

Using the approximation for small \( h \) where \( \ln(1 + h) \approx h \):

\[
\ln \left(\frac{2x + 3}{2x + 1}\right) \approx \ln \left(1 + \frac{2}{2x + 1}\right) \approx \frac{2}{2x + 1}
\]

Thus, 

\[
\ln y \approx (x + 1) \cdot \frac{2}{2x + 1}
\]

For large \( x \):

\[
\frac{x + 1}{2x + 1} \approx \frac{x}{2x} = \frac{1}{2}
\]

So,

\[
\ln y \approx 2 \cdot \frac{1}{2} = 1
\]

Hence,

\[
y \approx e
\]

Therefore, the limit for the second problem is:
\[
e
\]

### Summary of Answers:
1. \(\infty\)
2. \(e\)

Would you like more details or have any questions about these solutions?

Here are some additional questions you might consider:
1. How do you evaluate limits involving indeterminate forms?
2. What techniques can be used to simplify complex expressions inside limits?
3. How do exponential limits with bases approaching 1 behave?
4. What are some common approximations used in limit calculations?
5. How can you apply the natural logarithm to simplify limit problems?

**Tip:** When solving limits at infinity, always consider the dominant terms in the expressions to simplify your calculations.

Explore detailed step-by-step solutions for limit problems at infinity. Learn techniques to evaluate limits involving indeterminate forms and exponential limits. Includes analysis of complex expressions and applications of natural logarithms. Suitable for advanced high school students and college-level learners.

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