Use the properties of limits to help decide whether the limit exists. If the limit​ exists, find its value.
ModifyingBelow lim With x right arrow infinity StartFraction 7 x cubed minus 2 x minus 5 Over 6 x squared minus 3 x minus 8 EndFraction
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box within your choice.
A.ModifyingBelow lim With x right arrow infinity StartFraction 7 x cubed minus 2 x minus 5 Over 6 x squared minus 3 x minus 8 EndFraction equals seven sixths
ModifyingBelow lim With x right arrow infinity StartFraction 7 x cubed minus 2 x minus 5 Over 6 x squared minus 3 x minus 8 EndFraction
equalsseven sixths
B.
The limit does not exist and is neither infinity
 
nor minus
infinity.

To determine whether the limit exists, we can analyze the given expression as \(x \to \infty\) by using properties of limits, particularly focusing on the highest-degree terms in both the numerator and denominator.

The given expression is:

\[
\lim_{x \to \infty} \frac{7x^3 - 2x - 5}{6x^2 - 3x - 8}
\]

### Step 1: Identify the highest degree terms
- In the numerator, the highest-degree term is \(7x^3\).
- In the denominator, the highest-degree term is \(6x^2\).

### Step 2: Simplify the expression by dividing both the numerator and the denominator by the highest power of \(x\) in the denominator, which is \(x^2\):

\[
\lim_{x \to \infty} \frac{\frac{7x^3 - 2x - 5}{x^2}}{\frac{6x^2 - 3x - 8}{x^2}} = \lim_{x \to \infty} \frac{7x - \frac{2}{x} - \frac{5}{x^2}}{6 - \frac{3}{x} - \frac{8}{x^2}}
\]

### Step 3: Evaluate the limit as \(x \to \infty\)
As \(x \to \infty\):
- \(\frac{2}{x} \to 0\), \(\frac{5}{x^2} \to 0\), \(\frac{3}{x} \to 0\), and \(\frac{8}{x^2} \to 0\).

Thus, the limit simplifies to:

\[
\lim_{x \to \infty} \frac{7x - 0 - 0}{6 - 0 - 0} = \lim_{x \to \infty} \frac{7x}{6}
\]

### Step 4: Conclusion
As \(x \to \infty\), \(\frac{7x}{6} \to \infty\). Therefore, the limit does not exist because the function grows without bound as \(x \to \infty\).

Thus, the correct answer is:

**B. The limit does not exist and is neither infinity nor minus infinity.**

Would you like a detailed breakdown of these steps or have any questions? Here are some related questions for further understanding:

1. What are the key steps to simplifying limits at infinity involving polynomials?
2. How do lower-degree terms affect the evaluation of limits at infinity?
3. Can you explain why dividing by the highest degree in the denominator simplifies the problem?
4. How does one determine if a limit approaches infinity or negative infinity?
5. What happens to rational functions when the degree of the numerator is higher than the degree of the denominator?

**Tip:** Always focus on the highest-degree terms when evaluating limits as \(x \to \infty\) for rational functions.

Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. lim (x → ∞) (7x^3 - 2x - 5) / (6x^2 - 3x - 8)

Learn how to find the limit of the rational function (7x^3 - 2x - 5) / (6x^2 - 3x - 8) as x approaches infinity using polynomial division and properties of limits. This step-by-step solution explains key concepts in limits and rational functions, suitable for advanced high school students and college calculus learners.

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