Calculus Examples

Evaluate the Limit limit as n approaches 8 of n/(n^3+6n^2-12n-8i)
Step 1
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 2
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 3
Move the exponent from outside the limit using the Limits Power Rule.
Step 4
Move the term outside of the limit because it is constant with respect to .
Step 5
Move the exponent from outside the limit using the Limits Power Rule.
Step 6
Move the term outside of the limit because it is constant with respect to .
Step 7
Evaluate the limit of which is constant as approaches .
Step 8
Evaluate the limits by plugging in for all occurrences of .
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Step 8.1
Evaluate the limit of by plugging in for .
Step 8.2
Evaluate the limit of by plugging in for .
Step 8.3
Evaluate the limit of by plugging in for .
Step 8.4
Evaluate the limit of by plugging in for .
Step 9
Simplify the answer.
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Step 9.1
Cancel the common factor of and .
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Step 9.1.1
Factor out of .
Step 9.1.2
Cancel the common factors.
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Step 9.1.2.1
Factor out of .
Step 9.1.2.2
Factor out of .
Step 9.1.2.3
Factor out of .
Step 9.1.2.4
Factor out of .
Step 9.1.2.5
Factor out of .
Step 9.1.2.6
Factor out of .
Step 9.1.2.7
Factor out of .
Step 9.1.2.8
Cancel the common factor.
Step 9.1.2.9
Rewrite the expression.
Step 9.2
Multiply the numerator and denominator of by the conjugate of to make the denominator real.
Step 9.3
Multiply.
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Step 9.3.1
Combine.
Step 9.3.2
Multiply by .
Step 9.3.3
Simplify the denominator.
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Step 9.3.3.1
Expand using the FOIL Method.
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Step 9.3.3.1.1
Apply the distributive property.
Step 9.3.3.1.2
Apply the distributive property.
Step 9.3.3.1.3
Apply the distributive property.
Step 9.3.3.2
Simplify.
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Step 9.3.3.2.1
Multiply by .
Step 9.3.3.2.2
Multiply by .
Step 9.3.3.2.3
Raise to the power of .
Step 9.3.3.2.4
Raise to the power of .
Step 9.3.3.2.5
Use the power rule to combine exponents.
Step 9.3.3.2.6
Add and .
Step 9.3.3.2.7
Subtract from .
Step 9.3.3.2.8
Add and .
Step 9.3.3.3
Simplify each term.
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Step 9.3.3.3.1
Rewrite as .
Step 9.3.3.3.2
Multiply by .
Step 9.3.3.4
Add and .
Step 9.4
Split the fraction into two fractions.