7 lim x→+∞

−9 − 7

−8x

4 log3 x

=

A +∞

B −∞

C −9/4

D 0

+

E 0

−

Let's break this down step by step.

We are given the expression:

\[
\lim_{{x \to +\infty}} \left( \frac{-9 - 7}{-8x + 4 \log_3{x}} \right)
\]

Let's simplify the expression.

1. The numerator is a constant, \(-9 - 7 = -16\).
   So, the expression becomes:

   \[
   \lim_{{x \to +\infty}} \left( \frac{-16}{-8x + 4 \log_3{x}} \right)
   \]

2. Now, let's consider the denominator:

   - As \(x \to +\infty\), the term \(-8x\) dominates because it grows much faster than the logarithmic term \(4 \log_3{x}\).
   - Therefore, the denominator behaves like \(-8x\) as \(x\) becomes very large.

So, the expression simplifies to:

\[
\lim_{{x \to +\infty}} \left( \frac{-16}{-8x} \right)
\]

3. Simplifying this further:

\[
\lim_{{x \to +\infty}} \frac{-16}{-8x} = \lim_{{x \to +\infty}} \frac{2}{x}
\]

4. As \(x \to +\infty\), \(\frac{2}{x} \to 0\).

Thus, the limit of the entire expression is:

\[
0
\]

### Final answer: \( D \, 0 \)

Would you like more details or have any questions?

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Here are 5 related questions for practice:

1. What is \(\lim_{{x \to +\infty}} \frac{1}{x}\)?
2. What happens to the expression if the logarithmic term is stronger than the linear term?
3. How would the limit change if the coefficient of \(x\) in the denominator was positive?
4. What is \(\lim_{{x \to 0}} \frac{1}{x}\)?
5. How does the behavior of logarithmic functions compare to polynomial functions at infinity?

**Tip:** When evaluating limits at infinity, always identify which terms dominate in growth to simplify the expression.

7 lim x→+∞ −9 − 7 −8x 4 log3 x = A +∞ B −∞ C −9/4 D 0 + E 0 −

Explore the step-by-step solution to evaluate the limit of a rational expression involving logarithmic and linear terms as x approaches infinity. This problem highlights key mathematical concepts like limits, logarithmic growth, and asymptotic behavior. Suitable for advanced high school or early college students.

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