Math Problem Statement

7 lim x→+∞

−9 − 7

−8x

4 log3 x

=

A +∞

B −∞

C −9/4

D 0

E 0

Solution

Let's break this down step by step.

We are given the expression:

limx+(978x+4log3x)\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, 97=16-9 - 7 = -16. So, the expression becomes:

    limx+(168x+4log3x)\lim_{{x \to +\infty}} \left( \frac{-16}{-8x + 4 \log_3{x}} \right)

  2. Now, let's consider the denominator:

    • As x+x \to +\infty, the term 8x-8x dominates because it grows much faster than the logarithmic term 4log3x4 \log_3{x}.
    • Therefore, the denominator behaves like 8x-8x as xx becomes very large.

So, the expression simplifies to:

limx+(168x)\lim_{{x \to +\infty}} \left( \frac{-16}{-8x} \right)

  1. Simplifying this further:

limx+168x=limx+2x\lim_{{x \to +\infty}} \frac{-16}{-8x} = \lim_{{x \to +\infty}} \frac{2}{x}

  1. As x+x \to +\infty, 2x0\frac{2}{x} \to 0.

Thus, the limit of the entire expression is:

00

Final answer: D0D \, 0

Would you like more details or have any questions?


Here are 5 related questions for practice:

  1. What is limx+1x\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 xx in the denominator was positive?
  4. What is limx01x\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.

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Math Problem Analysis

Mathematical Concepts

Limits
Logarithmic functions
Asymptotic behavior

Formulas

Limit formula: lim x→∞ f(x) = L
Logarithmic growth: log_b{x} grows slower than linear functions

Theorems

Limit Laws
Logarithmic vs Polynomial Growth

Suitable Grade Level

Grades 11-12, Early College