# Evaluate limit as h approaches 0 of ((6+h)^-1-6^-1)/h

Take the limit of each term.
Simplify the limit argument.
Convert negative exponents to fractions.
Rewrite the expression using the negative exponent rule .
Rewrite the expression using the negative exponent rule .
Combine terms.
To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Multiply and .
Multiply and .
Reorder the factors of .
Combine the numerators over the common denominator.
Simplify the limit argument.
Multiply the numerator by the reciprocal of the denominator.
Multiply and .
Move the term outside of the limit because it is constant with respect to .
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 .
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 which is constant as approaches .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Simplify each term.
Multiply by .
Subtract from .
Evaluate the limit of the denominator.
Take the limit of each term.
Split the limit using the Product of Limits Rule on the limit as approaches .
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 which is constant as approaches .
Evaluate the limit of by plugging in for .
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
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 .
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 .
By the Sum Rule, the derivative of with respect to is .
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 .
Subtract from .
Differentiate using the Product Rule which states that is where and .
Differentiate using the Power Rule which states that is where .
Multiply by .
By the Sum Rule, the derivative of with respect to is .
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 .
Reorder terms.
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 .
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 .
Evaluate the limit of which is constant as approaches .
Simplify the denominator.
Multiply by .
Move the negative in front of the fraction.
Multiply .
Multiply and .
Multiply by .
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Evaluate limit as h approaches 0 of ((6+h)^-1-6^-1)/h