What is the limit as x approaches 1 of \sqrt[x-1]{x} ?

The limit as x approaches 1 of \sqrt[x-1]{x} is e

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limit as x approaches 1 of \sqrt[x-1]{x}

e

Decimal Notation

2.71828 \dots

Solution steps

Apply exponent rules

= x \to 1 lim (e^{\frac{1}{x - 1} l n (x)})

Simplify

e^{\frac{1}{x - 1} l n (x)} : x^{\frac{1}{x - 1}}

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

Apply exponent rules

= x \to 1 lim (e^{\frac{1}{x - 1} l n (x)})

Apply the Limit Chain Rule:

e

= e

x \to 3 + lim (\frac{1}{( x - 3 ) ^{2}})

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\frac{d y}{( 1 + 4 y ^{2} )} = - \frac{x}{( 1 + x ^{4} )} d x

6 x^{\frac{3}{2}}

x \to \infty lim (0.9 9^{x})

What is the limit as x approaches 1 of \sqrt[x-1]{x} ?
The limit as x approaches 1 of \sqrt[x-1]{x} is e