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.

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 .

Evaluate the limit of which is constant as approaches .

Combine the opposite terms in .

Add and .

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.

Differentiate the numerator and denominator.

By the Sum Rule, the derivative of with respect to is .

Evaluate .

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 .

By the Sum Rule, the derivative of with respect to is .

Since is constant with respect to , the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Add and .

Multiply by .

Since is constant with respect to , the derivative of with respect to is .

Add and .

Differentiate using the Power Rule which states that is where .

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 .

Move the limit inside the trig function because sine is continuous.

Split the limit using the Sum of Limits Rule on the limit as approaches .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of by plugging in for .

Evaluate the limit of which is constant as approaches .

Divide by .

Add and .

Evaluate ( limit as h approaches 0 of cos(x+h)-cos(x))/h