https://file.aiquickdraw.com/mathbot/file/1726208733011sd68x78r.jpg

The expression provided in the image is:

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

To solve this limit, let's follow these steps:

### Step 1: Direct Substitution
First, try substituting \( x = 3 \) into the expression to see if we get a determinate value or an indeterminate form.

Substituting \( x = 3 \):

\[
\text{Numerator: } \sqrt{2(3)^2 + 7} - 5 = \sqrt{18 + 7} - 5 = \sqrt{25} - 5 = 5 - 5 = 0
\]
\[
\text{Denominator: } 5(3) - 2 = 15 - 2 = 13
\]

So, the expression becomes:
\[
\lim_{{x \to 3}} \frac{0}{13} = 0
\]

### Conclusion:
Since the result of direct substitution is not an indeterminate form (e.g., \( \frac{0}{0} \)), the limit is simply \( 0 \).

The limit is:
\[
\boxed{0}
\]

Would you like any further details or clarifications?

Here are some related questions to extend this topic:
1. What is an indeterminate form, and how does it affect limit solving?
2. How would the limit change if the denominator approached zero instead of a constant?
3. Can L'Hôpital's Rule be applied in cases where direct substitution works?
4. What other techniques are available to solve limits besides direct substitution?
5. How does the limit behave for functions that involve radicals?

**Tip:** Always try direct substitution first when solving limits, as it often provides an immediate result if no indeterminate form is present.

Evaluate the limit: \( \lim_{{x \to 3}} \frac{\sqrt{2x^2 + 7} - 5}{5x - 2} \)

Learn how to solve the limit \( \lim_{{x \to 3}} \frac{\sqrt{2x^2 + 7} - 5}{5x - 2} \) using direct substitution. Explore algebraic techniques for handling limits involving radicals. Suitable for Grades 11-12.

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