Calculus Examples

$lim_{x \to 0} x (x^{x})$

Step 1

Multiply $x$ by $x^{x}$ .

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Step 1.1

Raise $x$ to the power of $1$ .

$lim_{x \to 0} x^{1} x^{x}$

Step 1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$lim_{x \to 0} x^{1 + x}$

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

Rewrite $x^{1 + x}$ as $e^{\ln (x^{1 + x})}$ .

$lim_{x \to 0} e^{\ln (x^{1 + x})}$

Step 2.2

Expand $\ln (x^{1 + x})$ by moving $1 + x$ outside the logarithm.

$lim_{x \to 0} e^{(1 + x) \ln (x)}$

Step 3

Because there are no values to the left of $0$ in the domain of $e^{(1 + x) \ln (x)}$ , the limit does not exist.

$Does not exist$