(e) $$ \lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x+2}{3 x^{2}-1} $$

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To evaluate the limit:

$$
\lim_{x \rightarrow \infty} \frac{5x^{2} - 3x + 2}{3x^{2} - 1}
$$

we can analyze the behavior of the function as $$ x $$ approaches infinity. Here's a step-by-step approach:

1. **Identify the Leading Terms:**
   In both the numerator and the denominator, the highest power of $$ x $$ is $$ x^2 $$.

- **Numerator:** $$ 5x^2 - 3x + 2 $$
   - **Denominator:** $$ 3x^2 - 1 $$

2. **Divide Every Term by $$ x^2 $$:**
   This simplifies the expression and makes it easier to evaluate the limit.

$$
   \frac{5x^{2} - 3x + 2}{3x^{2} - 1} = \frac{5 - \frac{3}{x} + \frac{2}{x^2}}{3 - \frac{1}{x^2}}
   $$

3. **Evaluate the Limit as $$ x $$ Approaches Infinity:**
   As $$ x $$ becomes very large, the terms with $$ \frac{1}{x} $$ and $$ \frac{1}{x^2} $$ approach zero.

$$
   \lim_{x \rightarrow \infty} \frac{5 - \frac{3}{x} + \frac{2}{x^2}}{3 - \frac{1}{x^2}} = \frac{5 - 0 + 0}{3 - 0} = \frac{5}{3}
   $$

4. **Conclusion:**
   The limit simplifies to the ratio of the leading coefficients of the highest power terms.

$$
   \lim_{x \rightarrow \infty} \frac{5x^{2} - 3x + 2}{3x^{2} - 1} = \frac{5}{3}
   $$

**Final Answer:**  
The limit is $$ \frac{5}{3} $$.
The limit is $$ \frac{5}{3} $$.

(e) \( \lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x+2}{3 x^{2}-1} \)

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