Math Problem Statement
lim_{x -> 1} ln(x/e^{8x})
Solution
We are tasked with evaluating the limit:
Step 1: Simplify the argument of the logarithm
First, rewrite the expression inside the logarithm:
Since (by the logarithmic identity ):
Thus, the limit becomes:
Step 2: Evaluate the limit
Now, evaluate each term at :
Therefore:
Conclusion:
The limit is:
Would you like further details on this solution or have any questions?
Follow-up questions:
- How does the natural logarithmic function behave as or ?
- Can you explain the properties of logarithms that were used in the simplification process?
- What is the significance of exponential growth in the expression ?
- How would this problem change if the exponent of was different?
- How do logarithms and exponentials interact in calculus and limits?
Tip:
When simplifying logarithmic expressions, always remember to apply logarithmic identities like and .
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Math Problem Analysis
Mathematical Concepts
Limits
Logarithmic Functions
Exponential Functions
Algebra
Formulas
ln(a/b) = ln(a) - ln(b)
lim_{x -> 1} ln(x) = ln(1) = 0
lim_{x -> 1} 8x = 8(1) = 8
Theorems
Logarithmic Identities
Properties of Limits
Suitable Grade Level
Grade 12 or Undergraduate Calculus