# Evaluate limit as x approaches 0 of cos(x)^(1/x)

Use the properties of logarithms to simplify the limit.
Rewrite as .
Expand by moving outside the logarithm.
Take the limit of each term.
Move the limit into the exponent.
Combine and .
Evaluate the limit of the numerator and the limit of the denominator.
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
Take the limit of each term.
Move the limit inside the logarithm.
Move the limit inside the trig function because cosine is continuous.
Evaluate the limit of by plugging in for .
The exact value of is .
The natural logarithm of is .
Evaluate the limit of by plugging in for .
The expression contains a division by The expression is undefined.
Undefined
Since is of indeterminate form, apply L’Hospital’s Rule. L’Hospital’s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Find the derivative of the numerator and denominator.
Differentiate the numerator and denominator.
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
Convert from to .
The derivative of with respect to is .
Remove parentheses.
Move to the left of .
Rewrite as .
Differentiate using the Power Rule which states that is where .
Take the limit of each term.
Split the limit using the Limits Quotient Rule on the limit as approaches .
Move the term outside of the limit because it is constant with respect to .
Split the limit using the Product of Limits Rule on the limit as approaches .
Move the limit inside the trig function because secant is continuous.
Move the limit inside the trig function because sine is continuous.
Evaluate the limits by plugging in for all occurrences of .
Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
Evaluate the limit of which is constant as approaches .