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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.

Determine the following limits: a) lim (x -> -∞) x / (x+1)^2, b) lim (x -> -∞) x^2 / (x^2 + 1), c) lim (x -> -∞) e^x / x, d) lim (x -> ∞) arctan(-x)e^x, e) lim (x -> ∞) e^(x/2), f) lim (x -> ∞) e^(2x) / x.

Solve a series of calculus limit problems covering concepts like exponential growth, polynomial functions, and arctangent limits. Detailed solutions include step-by-step application of limit laws, L'Hopital's Rule, and function behavior as x approaches infinity or negative infinity. Suitable for Grades 11-12 students learning calculus.

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