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 trig function because sine is continuous.

Move the term outside of the limit because it is constant with respect to .

Evaluate the limit of by plugging in for .

Simplify the answer.

Multiply by .

The exact value of is .

Evaluate the limit of the denominator.

Move the term outside of the limit because it is constant with respect to .

Evaluate the limit of by plugging in for .

Multiply by .

The expression contains a division by The expression is undefined.

Undefined

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.

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 .

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

Remove parentheses.

Move to the left of .

Multiply by .

Differentiate using the Power Rule which states that is where .

Multiply by .

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

Differentiate using the Power Rule which states that is where .

Multiply by .

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 cosine is continuous.

Move the term outside of the limit because it is constant with respect to .

Evaluate the limit of by plugging in for .

Evaluate the limit of which is constant as approaches .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Multiply by .

The exact value of is .

Evaluate limit as x approaches 0 of (sin(3x))/(3x)