What is the limit as x approaches 0+of (cos(x))^{1/(ln(x))} ?

The limit as x approaches 0+of (cos(x))^{1/(ln(x))} is 1

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limit as x approaches 0+of (cos(x))^{1/(ln(x))}

x \to 0 + lim ((cos (x))^{\frac{1}{l n ( x )}})

1

Solution steps

x \to 0 + lim ((cos (x))^{\frac{1}{l n ( x )}})

Apply exponent rules

= x \to 0 + lim (e^{\frac{1}{l n ( x )} l n (c o s (x))})

Apply the Limit Chain Rule:

1

= 1

x \to 1 lim (x^{9} - 1)

x \to 0 + lim (\frac{e ^{x}}{x} - \frac{1}{x})

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

x \to 0 + lim (x ln (1 + \frac{1}{x}))

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

What is the limit as x approaches 0+of (cos(x))^{1/(ln(x))} ?
The limit as x approaches 0+of (cos(x))^{1/(ln(x))} is 1