Evaluate limit as x approaches 0 of (1-4x)^(1/x)

Math
Take the limit of each term.
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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.
Combine and .
Apply L’Hospital’s rule.
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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.
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 .
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Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Simplify the answer.
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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.
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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 .
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 limits by plugging in for all occurrences of .
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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 answer.
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Simplify the denominator.
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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 (1-4x)^(1/x)

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