Question

Evaluate: \(\displaystyle \lim_{x \to 0^+} x^x\)

• A. \(0\)
• B. \(\infty\)
• C. \(e\)
• D. \(1\)

Hint:

Use logarithmic transformation for limits involving exponents like \(x^x\) or \(x^{1/x}\) for easier evaluation.

Answer 1

The Correct Option is D

Step 1: Take logarithm.
Let \(y = x^x\). Taking natural logarithm on both sides: \[ \ln y = x \ln x \]

Step 2: Evaluate limit.
As \(x \to 0^+\), \(\ln x \to -\infty\), so \(x \ln x \to 0\). \[ \lim_{x \to 0^+} \ln y = 0 \implies \ln y = 0 \] Hence \(y = e^0 = 1.\)

Step 3: Conclusion.
\[ \lim_{x \to 0^+} x^x = 1 \]