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I can help you solve each of the limits in the list you provided. Let's go through some of them step by step. I'll start with a few calculations, and we can proceed based on your preferences.

### Problem 1:
\[
\lim_{x \to 0} \frac{x^2 + 7x}{x}
\]

We can simplify the expression by factoring the numerator:
\[
\frac{x^2 + 7x}{x} = \frac{x(x + 7)}{x} = x + 7
\]
Now, taking the limit as \( x \to 0 \):
\[
\lim_{x \to 0} (x + 7) = 7
\]
**Answer**: \( 7 \)

### Problem 2:
\[
\lim_{x \to 0} \frac{x^2 - 5x}{x}
\]

Again, simplify the numerator:
\[
\frac{x^2 - 5x}{x} = \frac{x(x - 5)}{x} = x - 5
\]
Now, take the limit as \( x \to 0 \):
\[
\lim_{x \to 0} (x - 5) = -5
\]
**Answer**: \( -5 \)

### Problem 3:
\[
\lim_{x \to 2} \frac{x^2 + 5x + 6}{x + 2}
\]

Factor the numerator:
\[
x^2 + 5x + 6 = (x + 2)(x + 3)
\]
Thus:
\[
\frac{x^2 + 5x + 6}{x + 2} = \frac{(x + 2)(x + 3)}{x + 2} = x + 3 \quad \text{(for \( x \neq -2 \))}
\]
Now, take the limit as \( x \to 2 \):
\[
\lim_{x \to 2} (x + 3) = 5
\]
**Answer**: \( 5 \)

### Problem 4:
\[
\lim_{x \to 5} \frac{x^2 + 9x + 20}{5x - 2}
\]

First, plug \( x = 5 \) directly into the expression:
\[
\frac{5^2 + 9(5) + 20}{5(5) - 2} = \frac{25 + 45 + 20}{25 - 2} = \frac{90}{23}
\]
**Answer**: \( \frac{90}{23} \)

### Problem 5:
\[
\lim_{x \to 0} \frac{x^2 + 12x}{2x}
\]

Simplify the expression:
\[
\frac{x^2 + 12x}{2x} = \frac{x(x + 12)}{2x} = \frac{x + 12}{2}
\]
Now, take the limit as \( x \to 0 \):
\[
\lim_{x \to 0} \frac{x + 12}{2} = \frac{12}{2} = 6
\]
**Answer**: \( 6 \)

---

I can continue solving the remaining limits. Would you like me to proceed? Let me know if you'd like detailed steps for specific ones!

### Related Questions:
1. How do we handle limits that result in indeterminate forms like \( \frac{0}{0} \)?
2. What is L'Hôpital's Rule, and when is it applicable in limits?
3. How do we solve limits involving trigonometric functions?
4. Can limits always be computed directly, or are there alternative techniques such as substitution or factoring?
5. What are infinite limits, and how do we deal with limits as \( x \to \infty \)?

**Tip**: For rational functions, always check if direct substitution works first. If it leads to \( \frac{0}{0} \), consider factoring or using algebraic simplifications before applying advanced techniques like L'Hôpital's Rule.

List of limit problems including lim (x^2 + 7x)/x as x approaches 0, lim (x^2 - 5x)/x as x approaches 0, lim (x^2 + 5x + 6)/(x + 2) as x approaches 2, and more.

Step-by-step solutions for a variety of limit problems including algebraic simplifications, factoring techniques, and using L'Hopital's Rule. Learn how to calculate limits such as (x^2 + 7x)/x as x approaches 0 and more. Suitable for Grades 10-12.

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