To evaluate the limit of the expression $\lim_{x \to 3} \frac{5x + 2}{2x - 3}$, we follow these steps:

Step 1: Direct Substitution

First, attempt to substitute $x = 3$ directly into the expression:

$\frac{5(3) + 2}{2(3) - 3} = \frac{15 + 2}{6 - 3} = \frac{17}{3}$

Conclusion:

The limit as $x$ approaches 3 is:

$\lim_{x \to 3} \frac{5x + 2}{2x - 3} = \frac{17}{3}$

Would you like further details on any of the steps, or do you have any questions?

Related Questions:

1. What happens if the denominator approaches zero in similar limits?
2. How would the limit change if $x$ approaches a value other than 3?
3. What is the difference between evaluating limits at a point and at infinity?
4. Can you provide examples of limits that result in indeterminate forms like $0/0$?
5. How do we handle limits where the function has a discontinuity?

Tip: Always try direct substitution first when evaluating limits; if that leads to an indeterminate form, then explore algebraic simplifications or other techniques.