What is the limit as x approaches 0-of |sin(x)|^x ?

The limit as x approaches 0-of |sin(x)|^x is Does not exist

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limit as x approaches 0-of |sin(x)|^x

x \to 0 - lim (∣ sin (x) ∣^{x})

Does not exist

Solution steps

x \to 0 - lim (∣ sin (x) ∣^{x})

sin (x)

is negative when

x \to 0 -

. Therefore

∣ sin (x) ∣ = - sin (x)

= x \to 0 - lim ((- sin (x))^{x})

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

(- sin (x))^{x} = (- 1)^{x} (sin (x))^{x}

= x \to 0 - lim ((- 1)^{x} sin^{x} (x))

(- 1)^{x} sin^{x} (x)

undefined for

x

approaching

0

from the left

= Does not exist

What is the limit as x approaches 0-of |sin(x)|^x ?
The limit as x approaches 0-of |sin(x)|^x is Does not exist