# Evaluate ( limit as t approaches 0 of 1-cos(t))/t

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.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the limit inside the trig function because cosine is continuous.
Evaluate the limits by plugging in for all occurrences of .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Simplify each term.
The exact value of is .
Multiply by .
Subtract from .
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.
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
The derivative of with respect to is .
Multiply by .
Multiply by .
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 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 which is constant as approaches .