https://file.aiquickdraw.com/mathbot/file/1726410579979xwjhc2la.jpg

The image contains the following limit problem:

\[
\lim_{x \to 1} \frac{x - 1}{\sqrt{x^2 + 3x - 2}}
\]

Let's solve it step-by-step.

### Step 1: Evaluate the limit by direct substitution
If we substitute \(x = 1\) directly into the expression, we get:

\[
\frac{1 - 1}{\sqrt{1^2 + 3(1) - 2}} = \frac{0}{\sqrt{1 + 3 - 2}} = \frac{0}{\sqrt{2}} = 0
\]

However, this is not an indeterminate form, so we can directly conclude that:

\[
\lim_{x \to 1} \frac{x - 1}{\sqrt{x^2 + 3x - 2}} = 0
\]

### Final Answer: The limit equals **0**.

Would you like a detailed explanation on any part of the solution? 

Here are 5 related questions to deepen your understanding:

1. How do you determine when to use direct substitution in limits?
2. What is an indeterminate form in limits, and how do you recognize one?
3. What are some techniques for solving limits that result in indeterminate forms?
4. How do you differentiate between continuous and discontinuous functions when solving limits?
5. How can the squeeze theorem be useful in solving limit problems?

**Tip:** Always check if direct substitution works in limit problems before considering more complex methods like L'Hopital's rule or factoring.

lim (x -> 1) [(x - 1) / (sqrt(x^2 + 3x - 2))]

Understand how to solve the limit problem lim (x -> 1) [(x - 1) / sqrt(x^2 + 3x - 2)] using direct substitution. This step-by-step explanation is ideal for students learning advanced high school math concepts. The problem covers limits and square root functions.

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