# Evaluate limit as h approaches 0 of ( square root of 9+h-3)/h

Take the limit of each term.
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.
Multiply the argument of the limit by the conjugate.
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Combine the opposite terms in .
Reorder the factors in the terms and .
Subtract from .
Simplify each term.
Multiply .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite as .
Use to rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify.
Multiply by .
Combine the opposite terms in .
Subtract from .
Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
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 .
Use to rewrite as .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
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 .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Multiply by .
Move to the denominator using the negative exponent rule .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply the numerator by the reciprocal of the denominator.
Rewrite as .
Multiply by .
Move the term outside of the limit because it is constant with respect to .
Split the limit using the Limits Quotient Rule on the limit as approaches .
Move the limit under the radical sign.
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 the denominator.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply .
Multiply and .
Multiply by .
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Evaluate limit as h approaches 0 of ( square root of 9+h-3)/h

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