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:
Let's simplify the expression.
-
The numerator is a constant, . So, the expression becomes:
-
Now, let's consider the denominator:
- As , the term dominates because it grows much faster than the logarithmic term .
- Therefore, the denominator behaves like as becomes very large.
So, the expression simplifies to:
- Simplifying this further:
- As , .
Thus, the limit of the entire expression is:
Final answer:
Would you like more details or have any questions?
Here are 5 related questions for practice:
- What is ?
- What happens to the expression if the logarithmic term is stronger than the linear term?
- How would the limit change if the coefficient of in the denominator was positive?
- What is ?
- 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
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