7 lim x→+∞

−9 − 7

−8x

4 log3 x

=

A +∞

B −∞

C −9/4

D 0

+

E 0

−

We are tasked with finding the limit of the given expression as \( x \to +\infty \):

\[
\lim_{x \to +\infty} \frac{-9 - 7}{-8x} \cdot 4 \log_3 x
\]

Let’s simplify step by step.

### Step 1: Simplifying the numerator
The numerator is \( -9 - 7 = -16 \), so the expression becomes:

\[
\lim_{x \to +\infty} \frac{-16}{-8x} \cdot 4 \log_3 x
\]

### Step 2: Simplifying the fraction
\[
\frac{-16}{-8x} = \frac{16}{8x} = \frac{2}{x}
\]

Thus, the expression now is:

\[
\lim_{x \to +\infty} \frac{2}{x} \cdot 4 \log_3 x
\]

### Step 3: Simplifying the product
\[
\frac{2}{x} \cdot 4 \log_3 x = \frac{8 \log_3 x}{x}
\]

### Step 4: Evaluating the limit
As \( x \to +\infty \), we know the logarithmic term \( \log_3 x \) grows slowly, but the denominator \( x \) grows much faster than \( \log_3 x \). Therefore, the entire expression tends to 0 as \( x \to +\infty \).

### Conclusion:
The limit is \( 0 \).

Thus, the correct answer is:

\[
\boxed{E} \ 0+
\]

---

Would you like more details on any of the steps, or have any other questions?

Here are five related questions:
1. What is the behavior of logarithmic functions as \( x \to +\infty \)?
2. How do different types of functions grow compared to each other (polynomial, logarithmic, exponential)?
3. What is the limit of \( \frac{\log x}{x} \) as \( x \to +\infty \)?
4. What techniques are used to evaluate limits at infinity?
5. How does L'Hopital's Rule help in finding limits involving indeterminate forms?

**Tip**: When evaluating limits involving products, check which factors grow faster or slower to determine the overall behavior.

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

Learn how to solve the limit problem lim x→+∞ (−9 − 7)/(−8x) * 4 log3(x) = ?. This problem involves understanding logarithmic growth, polynomial terms, and limits at infinity. Ideal for students in Grades 11-12 and above, studying Precalculus or Calculus.

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