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

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To evaluate the limit:

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

**Step 1: Identify the Degrees of the Polynomial**

- **Numerator:** The highest power of $$ x $$ is $$ x^3 $$, so the degree is **3**.
- **Denominator:** The highest power of $$ x $$ is $$ x^5 $$, so the degree is **5**.

**Step 2: Compare the Degrees**

When evaluating limits of rational functions as $$ x $$ approaches infinity:

- If the **degree of the numerator** is **less than** the **degree of the denominator**, the limit is **0**.
- If the degrees are equal, the limit is the ratio of the leading coefficients.
- If the degree of the numerator is greater, the limit is either $$ \infty $$ or $$ -\infty $$, depending on the leading coefficients.

**Step 3: Apply to the Given Function**

In this case:
- Degree of numerator $$ = 3 $$
- Degree of denominator $$ = 5 $$

Since $$ 3 < 5 $$, the limit is:

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

**Answer:** 0
The limit is 0.

(b)

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