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 exponent from outside the limit using the Limits Power Rule.

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

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 .

Simplify the answer.

Simplify each term.

Raise to the power of .

Multiply by .

Subtract from .

Add and .

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 exponent from outside the limit using the Limits Power Rule.

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 .

Simplify the answer.

Raise to the power of .

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.

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

Differentiate using the Power Rule which states that is where .

Evaluate .

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 .

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

Add and .

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

Differentiate using the Power Rule which states that is where .

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

Add and .

Split the limit using the Limits Quotient Rule on the limit as approaches .

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

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

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

Move the exponent from outside the limit using the Limits Power Rule.

Evaluate the limit of by plugging in for .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of by plugging in for .

Cancel the common factor of and .

Factor out of .

Factor out of .

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Add and .

Raise to the power of .

Multiply by .

Divide by .

Evaluate limit as x approaches -3 of (x^2+6x+9)/(x^4-81)