Find $lim x \to 0 (| x |)^{x}$

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Solution

${lim}_{x \to 0} {| x |}^{x}$

$= {lim}_{x \to 0} {∣ ∣ e^{ln x} ∣ ∣}^{x}$

$= {lim}_{x \to 0} ∣ ∣ e^{x ln x} ∣ ∣$

$= e^{{lim}_{x \to 0} x ln x}$

$= e^{{lim}_{x \to 0} ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{ln x}{\frac{1}{x}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠}$

On applying differentiation on $⎛ ⎜ ⎜ ⎜ ⎝ \frac{ln x}{\frac{1}{x}} ⎞ ⎟ ⎟ ⎟ ⎠$ we get

$= e^{{lim}_{x \to 0} ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{\frac{1}{x}}{\frac{- 1}{x^{2}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠}$

$= e^{0}$

$= 1$

$\Rightarrow {lim}_{x \to 0} {| x |}^{x} = 1$

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