We are tasked with finding the limit:

$\lim_{{x \to 3}} \frac{3x^2 - 8}{x - 2}$

Step 1: Substitution to check if the limit can be evaluated directly.

First, substitute $x = 3$ into the function:

$\frac{3(3)^2 - 8}{3 - 2} = \frac{3(9) - 8}{1} = \frac{27 - 8}{1} = \frac{19}{1} = 19$

Since direct substitution does not result in any undefined form (like division by zero or $\frac{0}{0}$), the limit can be computed directly.

Conclusion:

$\lim_{{x \to 3}} \frac{3x^2 - 8}{x - 2} = 19$

Would you like a deeper explanation of any steps, or do you have further questions?

Related Questions:

1. What happens if the denominator was $x-3$ instead of $x-2$?
2. How would you evaluate the limit if direct substitution led to an indeterminate form?
3. Can this function have any vertical asymptotes?
4. How do you find the limit of more complex rational functions?
5. What is the general process for evaluating limits involving polynomials?

Tip:

Always check if direct substitution works when finding limits before using more advanced techniques.