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

Take the limit of each term.
Combine terms.
Write as a fraction with a common denominator.
Combine the numerators over the common denominator.
Use the properties of logarithms to simplify the limit.
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.
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
Take the limit of each term.
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 .
Cancel the common factor.
Divide by .
Cancel the common factor of .
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.
Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .
Divide by .
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.
Differentiate the numerator and denominator.
Differentiate using the chain rule, which states that is where and .
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 .
Multiply by .
Differentiate using the Power Rule which states that is where .
Multiply by .
Multiply and .
Rewrite the expression.
Simplify.
Apply the distributive property.
Apply the distributive property.
Simplify the numerator.
Subtract from .
Subtract from .
Multiply by .
Combine terms.
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
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.
Multiply by .
Multiply by .
Combine and .
Factor out of .
Factor out of .
Raise to the power of .
Factor out of .
Factor out of .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Cancel the common factor.
Rewrite the expression.
Take the limit of each term.
Divide the numerator and denominator by the highest power of in the denominator, which is .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Cancel the common factor of .
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.
Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .