Calculus Examples

Evaluate Using L'Hospital's Rule limit as x approaches 8 of (1-5/x)^x
Step 1
Combine terms.
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Step 1.1
Write as a fraction with a common denominator.
Step 1.2
Combine the numerators over the common denominator.
Step 2
Use the properties of logarithms to simplify the limit.
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Step 2.1
Rewrite as .
Step 2.2
Expand by moving outside the logarithm.
Step 3
Evaluate the limit.
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Step 3.1
Move the limit into the exponent.
Step 3.2
Split the limit using the Product of Limits Rule on the limit as approaches .
Step 3.3
Move the limit inside the logarithm.
Step 3.4
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 3.5
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 3.6
Evaluate the limit of which is constant as approaches .
Step 4
Evaluate the limits by plugging in for all occurrences of .
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Step 4.1
Evaluate the limit of by plugging in for .
Step 4.2
Evaluate the limit of by plugging in for .
Step 4.3
Evaluate the limit of by plugging in for .
Step 5
Simplify the answer.
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Step 5.1
Simplify by moving inside the logarithm.
Step 5.2
Exponentiation and log are inverse functions.
Step 5.3
Simplify the numerator.
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Step 5.3.1
Multiply by .
Step 5.3.2
Subtract from .
Step 5.4
Apply the product rule to .
Step 5.5
Raise to the power of .
Step 5.6
Raise to the power of .