What is the limit as x approaches 0+of |x|^x ?

The limit as x approaches 0+of |x|^x is 1

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limit as x approaches 0+of |x|^x

x \to 0 + lim (∣ x ∣^{x})

1

Solution steps

x \to 0 + lim (∣ x ∣^{x})

x

is positive when

x \to 0 +

. Therefore

∣ x ∣ = x

= x \to 0 + lim (x^{x})

Apply exponent rules

= x \to 0 + lim (e^{x l n (x)})

Apply the Limit Chain Rule:

1

= 1

x \to \infty lim (\frac{20}{x})

x \to \infty lim (x 2^{x})

x \to 1 - lim (2 x)

x \to 0 lim (2 x (ln (x))^{2})

x \to 0 + lim (ln (x) ln (x + 1))

What is the limit as x approaches 0+of |x|^x ?
The limit as x approaches 0+of |x|^x is 1