Calculus Examples

Evaluate the Limit limit as x approaches 9 of ( square root of x-3)/(x-9)
Step 1
Apply L'Hospital's rule.
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Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 1.1.1
Take the limit of the numerator and the limit of the denominator.
Step 1.1.2
Evaluate the limit of the numerator.
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Step 1.1.2.1
Evaluate the limit.
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Step 1.1.2.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.2.1.2
Move the limit under the radical sign.
Step 1.1.2.1.3
Evaluate the limit of which is constant as approaches .
Step 1.1.2.2
Evaluate the limit of by plugging in for .
Step 1.1.2.3
Simplify the answer.
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Step 1.1.2.3.1
Simplify each term.
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Step 1.1.2.3.1.1
Rewrite as .
Step 1.1.2.3.1.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.1.2.3.1.3
Multiply by .
Step 1.1.2.3.2
Subtract from .
Step 1.1.3
Evaluate the limit of the denominator.
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Step 1.1.3.1
Evaluate the limit.
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Step 1.1.3.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.3.1.2
Evaluate the limit of which is constant as approaches .
Step 1.1.3.2
Evaluate the limit of by plugging in for .
Step 1.1.3.3
Simplify the answer.
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Step 1.1.3.3.1
Multiply by .
Step 1.1.3.3.2
Subtract from .
Step 1.1.3.3.3
The expression contains a division by . The expression is undefined.
Undefined
Step 1.1.3.4
The expression contains a division by . The expression is undefined.
Undefined
Step 1.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 1.2
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.
Step 1.3
Find the derivative of the numerator and denominator.
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Step 1.3.1
Differentiate the numerator and denominator.
Step 1.3.2
By the Sum Rule, the derivative of with respect to is .
Step 1.3.3
Evaluate .
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Step 1.3.3.1
Use to rewrite as .
Step 1.3.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3.3
To write as a fraction with a common denominator, multiply by .
Step 1.3.3.4
Combine and .
Step 1.3.3.5
Combine the numerators over the common denominator.
Step 1.3.3.6
Simplify the numerator.
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Step 1.3.3.6.1
Multiply by .
Step 1.3.3.6.2
Subtract from .
Step 1.3.3.7
Move the negative in front of the fraction.
Step 1.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.5
Simplify.
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Step 1.3.5.1
Rewrite the expression using the negative exponent rule .
Step 1.3.5.2
Combine terms.
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Step 1.3.5.2.1
Multiply by .
Step 1.3.5.2.2
Add and .
Step 1.3.6
By the Sum Rule, the derivative of with respect to is .
Step 1.3.7
Differentiate using the Power Rule which states that is where .
Step 1.3.8
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.9
Add and .
Step 1.4
Multiply the numerator by the reciprocal of the denominator.
Step 1.5
Rewrite as .
Step 1.6
Multiply by .
Step 2
Evaluate the limit.
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Step 2.1
Move the term outside of the limit because it is constant with respect to .
Step 2.2
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 2.3
Evaluate the limit of which is constant as approaches .
Step 2.4
Move the limit under the radical sign.
Step 3
Evaluate the limit of by plugging in for .
Step 4
Simplify the answer.
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Step 4.1
Simplify the denominator.
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Step 4.1.1
Rewrite as .
Step 4.1.2
Pull terms out from under the radical, assuming positive real numbers.
Step 4.2
Multiply .
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Step 4.2.1
Multiply by .
Step 4.2.2
Multiply by .
Step 5
The result can be shown in multiple forms.
Exact Form:
Decimal Form: