Vault Of Euclid

Limit of (cos(1/x))^x as x approaches infinity

Limits core

Evaluate the limit: \[ \lim_{x\to+\infty} \left(\cos \frac{1}{x}\right)^{x} \]

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What you need first

Recognising indeterminate forms
Properties of logarithms
L'Hôpital's rule
The chain rule for differentiation
Derivatives of trigonometric and logarithmic functions

What this problem tests

Tests the ability to convert a $1^\infty$ indeterminate form into a $0/0$ form using logarithms, and to apply L'Hôpital's rule correctly.

This is a standard Calculus I problem testing the evaluation of exponential indeterminate forms.

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