Math Problem Statement
jak to się przekształciło krok po kroku
Solution
To clarify the transformation step by step, let's break it into parts:
Problem Setup
We have a limit of the form:
We aim to simplify the expression step by step.
Step 1: Simplify the fractions
The numerator and denominator are already fractions. Combine them as follows:
= \frac{2 \ln x}{x} \cdot \frac{-2}{x^{-\frac{3}{2}}}.$$ Here, multiplying by the reciprocal of the denominator simplifies the expression. --- ### Step 2: Simplify powers of $$x$$ Now combine the powers of $$x$$: 1. $$\frac{2 \ln x}{x}$$ stays as is. 2. $$\frac{-2}{x^{-\frac{3}{2}}}$$ simplifies as $$-2 x^{\frac{3}{2}}$$. Thus, multiplying the two terms: $$\frac{2 \ln x}{x} \cdot (-2 x^{\frac{3}{2}}) = -4 \ln x \cdot x^{\frac{3}{2} - 1}.$$ --- ### Step 3: Combine exponents Simplify the exponent of $$x$$: $$\frac{3}{2} - 1 = \frac{1}{2}.$$ So the expression becomes: $$-4 \ln x \cdot x^{\frac{1}{2}}.$$ --- ### Final Form We now have the simplified expression: $$\lim_{x \to 0^+} \frac{\frac{2 \ln x}{x}}{-\frac{1}{2} x^{-\frac{3}{2}}} = \lim_{x \to 0^+} -\frac{4 \ln x}{x^{-\frac{1}{2}}}.$$ --- ### Summary of Steps 1. Combine the fractions by multiplying by the reciprocal of the denominator. 2. Simplify the powers of $$x$$. 3. Combine exponents to reduce terms. 4. Final simplified form: $$-\frac{4 \ln x}{x^{-\frac{1}{2}}}$$. Let me know if you'd like further clarification on any step! --- ### Follow-up Questions 1. How do you solve limits involving logarithmic functions approaching $$0^+$$? 2. Can you simplify other similar expressions involving negative exponents? 3. How do you interpret the behavior of $$x^{\frac{1}{2}}$$ as $$x \to 0^+$$? 4. What role does the negative sign play in this transformation? 5. What happens if the base $$x$$ approaches infinity instead of zero? ### Tip When simplifying limits with fractions, always combine terms carefully and simplify powers of $$x$$ step by step.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Logarithmic Functions
Exponential Rules
Simplification of Expressions
Formulas
Logarithmic Limit: lim(x→0^+) ln(x)
Fraction Simplification: a/b ÷ c/d = a/b × d/c
Exponent Rules: x^a * x^b = x^(a+b)
Theorems
Laws of Exponents
Logarithmic Simplification
Suitable Grade Level
High School - Advanced (Grades 11-12)
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