# Evaluate ( limit as x approaches 0 of cos(9x)-1)/(x^2)

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 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 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 each term.
Multiply by .
The exact value of is .
Subtract from .
Evaluate the limit of the denominator.
Move the exponent from outside the limit using the Limits Power Rule.
Evaluate the limit of by plugging in for .
Raising to any positive power yields .
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 .
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 .
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 .
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 .
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.
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.
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 exact value of is .
Multiply by .
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.
Find the derivative of the numerator and denominator.
Differentiate the numerator and denominator.
Since is constant with respect to , the derivative of with respect to is .
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 .
Remove parentheses.
Since is constant with respect to , the derivative of with respect to is .
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 .
Take the limit of each term.
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 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 numerator.
Multiply by .
The exact value of is .
Multiply by .
Move the negative in front of the fraction.
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Mixed Number Form:
Evaluate ( limit as x approaches 0 of cos(9x)-1)/(x^2)