What is the limit as x approaches 0+of (3x)^{x^2} ?

The limit as x approaches 0+of (3x)^{x^2} is 1

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limit as x approaches 0+of (3x)^{x^2}

x \to 0 + lim ((3 x)^{x^{2}})

1

Solution steps

x \to 0 + lim ((3 x)^{x^{2}})

Simplify

(3 x)^{x^{2}} : 3^{x^{2}} x^{x^{2}}

= x \to 0 + lim (3^{x^{2}} x^{x^{2}})

x \to a lim [f (x) \cdot g (x)] = x \to a lim f (x) \cdot x \to a lim g (x)

With the exception of indeterminate form

= x \to 0 + lim (3^{x^{2}}) \cdot x \to 0 + lim (x^{x^{2}})

x \to 0 + lim (3^{x^{2}}) = 1

x \to 0 + lim (x^{x^{2}}) = 1

= 1 \cdot 1

Simplify

= 1

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What is the limit as x approaches 0+of (3x)^{x^2} ?
The limit as x approaches 0+of (3x)^{x^2} is 1