Take the limit of the numerator and the limit of the denominator.

Evaluate the limit of the numerator.

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

Evaluate the limit of by plugging in for .

The exact value of is .

Evaluate the limit of the denominator.

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 the answer.

Simplify each term.

The exact value of is .

Multiply by .

Subtract from .

The expression contains a division by The expression is undefined.

Undefined

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.

The derivative of with respect to is .

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 .

Add and .

Convert from to .

Consider the left sided limit.

As the values approach from the left, the function values decrease without bound.

Consider the right sided limit.

As the values approach from the right, the function values increase without bound.

Since the left sided and right sided limits are not equal, the limit does not exist.

Evaluate limit as x approaches 0 of (sin(x))/(1-cos(x))