Finding the limit of x^x as x approaches 0 from the right involves rewriting the expression using exponential form, applying L’Hôpital’s Rule, and simplifying to get the result 1.
Finding the Limit of \(x^x\) as \(x\) Approaches 0 from the Right
Start with the Given Limit: We begin with the expression \(\lim_{{x \rightarrow 0^+}} x^x\), where we want to find the limit as \(x\) approaches 0 from the right.
Express the Function Using Exponential Form: We rewrite the expression using the property \(x^x = e^{x \ln(x)}\), resulting in \(\lim_{{x \rightarrow 0^+}} e^{x \ln(x)}\).
Move the Limit Inside the Exponent: We move the limit inside the exponent, giving us \(e^{\lim_{{x \rightarrow 0^+}} x \ln(x)}\).
Recognize the Indeterminate Form \(0^0\): We notice that the expression inside the limit is in the indeterminate form \(0^0\).
Rewrite the Expression Inside the Limit: We rewrite the expression inside the limit as \(\lim_{{x \rightarrow 0^+}} \left( \ln(x) / \left[ 1/x \right] \right)\) to achieve the \(-\infty/\infty\) form suitable for L’Hôpital’s Rule.
Apply L’Hôpital’s Rule: We apply L’Hôpital’s Rule to the expression inside the limit, giving us \(e^{\lim_{{x \rightarrow 0^+}} \frac{1/x}{-1/x^2}}\).
Simplify the Expression: We simplify the expression inside the limit by multiplying the numerator and denominator by \(x^2\), resulting in \(e^{\lim_{{x \rightarrow 0^+}} \left( \frac{1}{x} \cdot \left( -x^2 \right) \right)}\).
Simplify Further: We further simplify the expression inside the limit by multiplying \(-x^2\) with \(1/x\), giving us \(e^{\lim_{{x \rightarrow 0^+}} \left( -x \right)}\).
Evaluate the Limit: We evaluate the limit of \(-x\) as \(x\) approaches 0 from the right, resulting in \(e^{-0}\).
Final Result: Since \(e^{-0} = 1\), the final result of the limit is \(1\).
