We are asked to evaluate the limit as $x$ approaches 3 for the following expression:

$\lim_{x \to 3} \frac{10x^3 + 3x^2 - 2x}{15x^3 + x^2 + 2x + 1}$

Step 1: Plugging in $x = 3$

Substitute $x = 3$ into both the numerator and the denominator.

For the numerator:

$10(3)^3 + 3(3)^2 - 2(3) = 10(27) + 3(9) - 6 = 270 + 27 - 6 = 291$

For the denominator:

$15(3)^3 + (3)^2 + 2(3) + 1 = 15(27) + 9 + 6 + 1 = 405 + 9 + 6 + 1 = 421$

Step 2: Forming the fraction

Now, the expression becomes:

$\frac{291}{421}$

Thus, the limit as $x$ approaches 3 is:

$\boxed{\frac{291}{421}}$

Would you like further details on any part of the solution?

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Here are 5 related questions to deepen your understanding:

1. What happens if you change the limit point from 3 to infinity in the same expression?
2. How can L'Hopital's Rule be used to solve limits that result in indeterminate forms?
3. What if the degree of the numerator is less than that of the denominator, how would the limit behave as $x \to \infty$?
4. Can you calculate the limit for other polynomial expressions at different values?
5. How does the behavior of the expression change for values of $x$ close to 0?

Tip: Always try substituting the limit value directly into the expression first. If the result is not indeterminate, this is the fastest way to solve the limit.