# Evaluate limit as x approaches nine of (x to the power of two minus eighty-one) divide(x minus nine)

Apply L'Hospital's rule.

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 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 .

Evaluate the limit of the denominator.

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

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 .

Subtract from .

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.

Find the derivative of the numerator and denominator.

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 .

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 .

Divide by .

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

Evaluate the limit of by plugging in for .

Multiply by .

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