Evaluate limit as x approaches infinity of (1+1/x)^x

Math
Take the limit of each term.
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Combine terms.
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Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Use the properties of logarithms to simplify the limit.
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Rewrite as .
Expand by moving outside the logarithm.
Move the limit into the exponent.
Rewrite as .
Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
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Take the limit of each term.
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Move the limit inside the logarithm.
Divide the numerator and denominator by the highest power of in the denominator, which is .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
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 .
Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .
Evaluate the limits by plugging in the value for the variable.
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Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .
Simplify the answer.
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Divide by .
Add and .
The natural logarithm of is .
Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .
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.
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Differentiate the numerator and denominator.
Differentiate using the chain rule, which states that is where and .
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To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
Multiply by the reciprocal of the fraction to divide by .
Multiply by .
Differentiate using the Quotient Rule which states that is where and .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Add and .
Multiply by .
Differentiate using the Power Rule which states that is where .
Multiply by .
Multiply and .
Rewrite the expression.
Simplify.
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Apply the distributive property.
Apply the distributive property.
Simplify the numerator.
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Subtract from .
Subtract from .
Multiply by .
Combine terms.
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Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Multiply by .
Move the negative in front of the fraction.
Rewrite as .
Differentiate using the Power Rule which states that is where .
Rewrite the expression using the negative exponent rule .
Combine factors.
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Multiply by .
Multiply by .
Combine and .
Factor out of .
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Factor out of .
Raise to the power of .
Factor out of .
Factor out of .
Cancel the common factor of and .
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Factor out of .
Cancel the common factors.
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Cancel the common factor.
Rewrite the expression.
Take the limit of each term.
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Divide the numerator and denominator by the highest power of in the denominator, which is .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
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 .
Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .
Evaluate the limits by plugging in the value for the variable.
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Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .
Simplify the answer.
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Add and .
Divide by .
Simplify.
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Evaluate limit as x approaches infinity of (1+1/x)^x

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