Math Problem Statement

lim_{x -> 1} ln(x/e^{8x})

Solution

We are tasked with evaluating the limit:

limx1ln(xe8x)\lim_{{x \to 1}} \ln\left( \frac{x}{e^{8x}} \right)

Step 1: Simplify the argument of the logarithm

First, rewrite the expression inside the logarithm:

ln(xe8x)=ln(x)ln(e8x)\ln\left( \frac{x}{e^{8x}} \right) = \ln(x) - \ln(e^{8x})

Since ln(e8x)=8x\ln(e^{8x}) = 8x (by the logarithmic identity ln(ea)=a\ln(e^a) = a):

ln(x)8x\ln(x) - 8x

Thus, the limit becomes:

limx1(ln(x)8x)\lim_{{x \to 1}} \left( \ln(x) - 8x \right)

Step 2: Evaluate the limit

Now, evaluate each term at x=1x = 1:

  • limx1ln(x)=ln(1)=0\lim_{{x \to 1}} \ln(x) = \ln(1) = 0
  • limx18x=8(1)=8\lim_{{x \to 1}} 8x = 8(1) = 8

Therefore:

limx1(ln(x)8x)=08=8\lim_{{x \to 1}} \left( \ln(x) - 8x \right) = 0 - 8 = -8

Conclusion:

The limit is:

8\boxed{-8}

Would you like further details on this solution or have any questions?

Follow-up questions:

  1. How does the natural logarithmic function behave as x0x \to 0 or xx \to \infty?
  2. Can you explain the properties of logarithms that were used in the simplification process?
  3. What is the significance of exponential growth in the expression e8xe^{8x}?
  4. How would this problem change if the exponent of ee was different?
  5. How do logarithms and exponentials interact in calculus and limits?

Tip:

When simplifying logarithmic expressions, always remember to apply logarithmic identities like ln(ab)=bln(a)\ln(a^b) = b \ln(a) and ln(ab)=ln(a)+ln(b)\ln(ab) = \ln(a) + \ln(b).

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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