Let's solve the given limits one by one.

a) $\lim_{x \to -\infty} \frac{x}{(x + 1)^2}$

As $x \to -\infty$:

• The numerator is $x \to -\infty$.
• The denominator is $(x + 1)^2 \approx x^2$ (since adding 1 is negligible as $x$ becomes very large).

Thus, the expression becomes approximately:

$\frac{x}{x^2} = \frac{1}{x} \to 0 \quad \text{as} \quad x \to -\infty$

So the limit is:

$\boxed{0}$

b) $\lim_{x \to -\infty} \frac{x^2}{x^2 + 1}$

As $x \to -\infty$:

• The numerator $x^2$ grows very large.
• The denominator $x^2 + 1 \approx x^2$ (since 1 becomes negligible).

Thus, the expression becomes approximately:

$\frac{x^2}{x^2 + 1} \approx \frac{x^2}{x^2} = 1$

So the limit is:

$\boxed{1}$

c) $\lim_{x \to -\infty} \frac{e^x}{x}$

As $x \to -\infty$:

• The exponential function $e^x$ approaches 0 very quickly.
• The linear term $x$ approaches $-\infty$.

Thus, we are dividing a very small number by a very large negative number, so the expression tends to:

$0$

So the limit is:

$\boxed{0}$

d) $\lim_{x \to \infty} \arctan(-x) e^x$

As $x \to \infty$:

• $\arctan(-x) \to -\frac{\pi}{2}$.
• $e^x$ grows exponentially fast to $\infty$.

Thus, the product is:

$-\frac{\pi}{2} \cdot \infty = -\infty$

So the limit is:

$\boxed{-\infty}$

e) $\lim_{x \to \infty} e^{x/2}$

As $x \to \infty$, the expression $e^{x/2}$ grows without bound since it is an exponential function. Therefore:

$e^{x/2} \to \infty$

So the limit is:

$\boxed{\infty}$

f) $\lim_{x \to \infty} \frac{e^{2x}}{x}$

As $x \to \infty$:

• $e^{2x}$ grows exponentially fast.
• $x$ grows linearly.

Thus, the exponential growth dominates, and the expression tends to:

$\frac{\infty}{x} = \infty$

So the limit is:

$\boxed{\infty}$

Would you like further details on any of these solutions? Here are five related questions you could explore:

1. What is the behavior of exponential functions as $x \to \pm\infty$?
2. How can we use L'Hopital's Rule to evaluate indeterminate forms?
3. Why do terms like $x + 1$ become negligible for large values of $x$?
4. What is the significance of the $\arctan(x)$ function's limits?
5. How can the growth rates of different functions (e.g., exponential vs. polynomial) be compared?

Tip: For large $x$, the term that grows fastest often dictates the behavior of the whole expression.