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.