Use the properties of logarithms to simplify the limit.

Rewrite as .

Expand by moving outside the logarithm.

Move the limit into the exponent.

Combine and .

Apply L’Hospital’s rule.

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.

Split the limit using the Sum of Limits Rule on the limit as approaches .

Move the term outside of the limit because it is constant with respect to .

Evaluate the limits by plugging in for all occurrences of .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of by plugging in for .

Simplify the answer.

Multiply by .

Add and .

The natural logarithm of is .

Evaluate the limit of by plugging in for .

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 .

By the Sum Rule, the derivative of with respect to is .

Since is constant with respect to , the derivative of with respect to is .

Add and .

Since is constant with respect to , the derivative of with respect to is .

Combine and .

Move the negative in front of the fraction.

Differentiate using the Power Rule which states that is where .

Multiply by .

Differentiate using the Power Rule which states that is where .

Multiply the numerator by the reciprocal of the denominator.

Multiply by .

Move the term outside of the limit because it is constant with respect to .

Move the term outside of the limit because it is constant with respect to .

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 .

Move the term outside of the limit because it is constant with respect to .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of by plugging in for .

Simplify the denominator.

Multiply by .

Add and .

Divide by .

Multiply by .

Rewrite the expression using the negative exponent rule .

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Evaluate limit as x approaches 0 of (1-2x)^(1/x)