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Let's go through a few of these limit problems one by one.

### Problem 19:
\[
\lim_{x \to 1} \frac{\sqrt{x} - 1}{x - 1}
\]
To solve this limit, notice that directly substituting \(x = 1\) leads to a \( \frac{0}{0} \) indeterminate form. So, we can rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator:

\[
\frac{\sqrt{x} - 1}{x - 1} \times \frac{\sqrt{x} + 1}{\sqrt{x} + 1} = \frac{x - 1}{(x - 1)(\sqrt{x} + 1)} = \frac{1}{\sqrt{x} + 1}
\]

Now, as \(x \to 1\), we get:

\[
\frac{1}{\sqrt{1} + 1} = \frac{1}{1 + 1} = \frac{1}{2}
\]

Thus, the limit is:
\[
\boxed{\frac{1}{2}}
\]

### Problem 20:
\[
\lim_{x \to 1} \frac{x^3 - 1}{x^2 - 1}
\]
Again, substituting \(x = 1\) gives a \( \frac{0}{0} \) indeterminate form. We can factor both the numerator and denominator:

\[
x^3 - 1 = (x - 1)(x^2 + x + 1)
\]
\[
x^2 - 1 = (x - 1)(x + 1)
\]

Cancel out the common factor \(x - 1\):

\[
\lim_{x \to 1} \frac{(x - 1)(x^2 + x + 1)}{(x - 1)(x + 1)} = \lim_{x \to 1} \frac{x^2 + x + 1}{x + 1}
\]

Now, substituting \(x = 1\):

\[
\frac{1^2 + 1 + 1}{1 + 1} = \frac{3}{2}
\]

Thus, the limit is:
\[
\boxed{\frac{3}{2}}
\]

Would you like me to continue solving the remaining problems? Let me know if you'd like any specific details or more problems explained!

Here are some follow-up questions:
1. How can limits help in understanding the behavior of functions near points of discontinuity?
2. What are common techniques to resolve indeterminate forms like \( \frac{0}{0} \)?
3. How do conjugates work in rationalizing expressions?
4. Why is factoring an effective method in solving limit problems?
5. What are other forms of indeterminate expressions besides \( \frac{0}{0} \)?

**Tip**: Always check for common factors in the numerator and denominator when dealing with rational expressions before applying limits.

Solve the following limit problems:
19. \( \lim_{x \to 1} \frac{\sqrt{x} - 1}{x - 1} \)
20. \( \lim_{x \to 1} \frac{x^3 - 1}{x^2 - 1} \)
21. \( \lim_{x \to 2} \frac{x^5 - 32}{x^3 - 8} \)
22. \( \lim_{x \to 1} \frac{x^{1/3} - 1}{x^{1/2} - 1} \)
23. \( \lim_{x \to 1} \frac{x^5 - x^{-5}}{x - 1} \)
24. \( \lim_{x \to 1} \frac{x^9 - x^{-9}}{x^7 - x^{-7}} \)
25. \( \lim_{x \to 3} \frac{x^4 - 81}{x^3 + 27} \)
26. \( \lim_{x \to 1} \frac{x^{3/4} - 1}{1 - x^{4/3}} \)
27. \( \lim_{x \to 3} \frac{\sqrt{3x} - 3}{x^2 - 9} \)

This guide provides step-by-step solutions to common limit problems involving algebraic manipulation, factoring, and rationalization. Learn how to tackle indeterminate forms such as 0/0 and apply key concepts like the difference of squares and cube roots. Suitable for high school students in Grades 10-12.

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