https://file.aiquickdraw.com/mathbot/file/1726759001922r9eqygcj.jpg

The expression in the image is:

\[
\lim_{x \to 3} \sqrt{5x^2 - 2x + 7}
\]

To solve this, we will evaluate the limit as \(x\) approaches 3 by directly substituting \(x = 3\) into the expression inside the square root.

### Substituting \(x = 3\) into the expression:

\[
5(3)^2 - 2(3) + 7 = 5(9) - 6 + 7 = 45 - 6 + 7 = 46
\]

Now, the limit becomes:

\[
\lim_{x \to 3} \sqrt{46} = \sqrt{46}
\]

Thus, the value of the limit is:

\[
\sqrt{46}
\]

Let me know if you want further details or if you have any questions!

Here are 5 relative questions you might consider:

1. What happens if \(x\) approaches a different number in this limit?
2. How do we handle limits that result in indeterminate forms, like \( \frac{0}{0} \)?
3. Can limits involving square roots result in negative numbers?
4. How does L'Hopital's rule apply to limits of more complex expressions?
5. How do you approach limits at infinity?

**Tip:** When evaluating limits, always check first if direct substitution works before applying more advanced techniques.

Evaluate the limit as x approaches 3 for the expression \( \lim_{x \to 3} \sqrt{5x^2 - 2x + 7} \).

Learn how to calculate the limit of the square root expression \( \lim_{x \to 3} \sqrt{5x^2 - 2x + 7} \) by using direct substitution. This step-by-step solution demonstrates the limit calculation process suitable for high school students.

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