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### 7. End Behavior for the Function:
\[ f(x) = \frac{3x^2 + 5x + 6}{x^2 - 4} \]

To determine the end behavior as \( x \to +\infty \) and \( x \to -\infty \), we focus on the highest-degree terms in the numerator and denominator.

- As \( x \to \infty \):
  \[
  f(x) \approx \frac{3x^2}{x^2} = 3
  \]
  Therefore, \( \lim_{x \to +\infty} f(x) = 3 \).

- As \( x \to -\infty \):
  The same leading terms dominate:
  \[
  f(x) \approx \frac{3x^2}{x^2} = 3
  \]
  Therefore, \( \lim_{x \to -\infty} f(x) = 3 \).

Thus, the end behavior of this function tends to 3 as \( x \to \pm \infty \).

---

### 8. End Behavior for the Function:
\[ f(x) = \frac{10 + 5x^2 - 3x^3}{2x^3 - 4x + 12} \]

Again, we'll focus on the highest-degree terms in both the numerator and the denominator.

- As \( x \to +\infty \):
  The highest degree term in both the numerator and denominator is \( -3x^3 \) and \( 2x^3 \) respectively:
  \[
  f(x) \approx \frac{-3x^3}{2x^3} = -\frac{3}{2}
  \]
  So, \( \lim_{x \to +\infty} f(x) = -\frac{3}{2} \).

- As \( x \to -\infty \):
  The leading terms still dominate, so the end behavior is the same:
  \[
  f(x) \approx \frac{-3x^3}{2x^3} = -\frac{3}{2}
  \]
  So, \( \lim_{x \to -\infty} f(x) = -\frac{3}{2} \).

Therefore, the function tends to \( -\frac{3}{2} \) as \( x \to \pm \infty \).

---

### 10. End Behavior for the Function:
\[ f(x) = e^x \]

For exponential functions like \( e^x \):

- As \( x \to +\infty \), \( f(x) = e^x \to \infty \).
- As \( x \to -\infty \), \( f(x) = e^x \to 0 \).

Therefore:
- \( \lim_{x \to +\infty} e^x = \infty \)
- \( \lim_{x \to -\infty} e^x = 0 \).

---

### 62. Horizontal and Vertical Asymptotes for the Rational Function:
\[ f(x) = \frac{5x - 2}{2x + 3} \]

**Horizontal Asymptotes**:
We check the degrees of the numerator and denominator.

- Since both the numerator and denominator are linear (degree 1), the horizontal asymptote is the ratio of the leading coefficients:
  \[
  \lim_{x \to \pm \infty} f(x) = \frac{5}{2}
  \]
  Therefore, the horizontal asymptote is \( y = \frac{5}{2} \).

**Vertical Asymptotes**:
We find vertical asymptotes by setting the denominator equal to zero:
\[
2x + 3 = 0 \quad \Rightarrow \quad x = -\frac{3}{2}
\]
Thus, the vertical asymptote is \( x = -\frac{3}{2} \).

---

Would you like any further details or have any questions on these?

Here are 5 additional related questions:
1. What is the difference between horizontal and oblique asymptotes?
2. How do we find the slant asymptote of a rational function?
3. What happens to the end behavior of a rational function when the degree of the numerator is greater than that of the denominator?
4. How can we confirm vertical asymptotes using limits?
5. Why does \( e^x \) tend to zero as \( x \to -\infty \)?

**Tip**: When dealing with rational functions, always check if the degree of the numerator is greater than the degree of the denominator to identify whether there is a slant or horizontal asymptote.

Determine the end behavior of the given functions and find all horizontal and vertical asymptotes for the rational function.

Learn how to determine the end behavior of rational and exponential functions, find horizontal and vertical asymptotes, and analyze the functions as x approaches positive and negative infinity. This guide covers detailed step-by-step solutions, focusing on key calculus and algebra concepts.

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