https://file.aiquickdraw.com/mathbot/file/1727455479403bln3gxlh.jpg

The image shows the limit problem:

\[
\lim_{{x \to 1}} (x^2 + 5x + 2) = 6
\]

Let's solve it step by step.

### Step 1: Evaluate the limit directly by substitution.
We can plug \(x = 1\) into the expression \(x^2 + 5x + 2\).

\[
1^2 + 5(1) + 2 = 1 + 5 + 2 = 8
\]

### Step 2: Compare the result with the given value.
The limit we calculated, 8, is not equal to 6, as the equation suggests.

This implies that there might be a mistake in either the problem or the way it's presented. Given the result doesn't match the provided solution, the limit seems incorrect based on the function given.

Would you like more details on this or have questions about this result?

Here are 5 related questions for deeper understanding:
1. How do you handle limits when direct substitution results in an undefined form like \(\frac{0}{0}\)?
2. Can limits always be evaluated by direct substitution?
3. What are the different types of indeterminate forms in limits?
4. How can the Squeeze Theorem be used to evaluate certain limits?
5. What are some alternative techniques for solving limits when direct substitution doesn't work?

**Tip:** Always check if direct substitution is possible before exploring other techniques like factoring or L'Hopital's rule.

Learn how to evaluate the limit of (x^2 + 5x + 2) as x approaches 1. This problem highlights key concepts in limit evaluation, substitution method, and provides a detailed step-by-step solution. Suitable for students in Grades 10-12.

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