Evaluate Using L'Hospital's Rule limit as x approaches 0 of (1-2x)^(1/x)

Problem

(lim_x→0)(1−2*x)

Solution

1. Identify the indeterminate form by substituting x=0 into the expression, which yields 1

2. Rewrite the limit using the natural logarithm to transform the power into a product, setting y=(1−2*x)1/x

3. Apply the natural logarithm to both sides to get ln(y)=1/x*ln(1−2*x) which can be written as a fraction.

4. Evaluate the limit of the natural logarithm as x approaches 0

(lim_x→0)(ln(1−2*x)/x)

5. Recognize that this results in the indeterminate form 0/0 allowing the use of L'Hospital's Rule.

6. Differentiate the numerator and the denominator with respect to x

d(ln(1−2*x))/d(x)=(−2)/(1−2*x)

d(x)/d(x)=1

7. Calculate the new limit using the derivatives.

(lim_x→0)((−2)/(1−2*x)/1)=−2

8. Exponentiate the result to find the original limit, since ln(L)=−2⇒L=e(−2)

Final Answer

(lim_x→0)(1−2*x)=e(−2)

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