Evaluate the Limit 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 expression using the natural logarithm and the exponential function to move the variable out of the exponent.

y=(1−2*x)^1/x

ln(y)=ln((1−2*x)^1/x)

ln(y)=1/x*ln(1−2*x)

3. Evaluate the limit of the logarithm as x approaches 0

(lim_x→0)(ln(1−2*x)/x)

4. Apply L'Hôpital's Rule because the limit is in the 0/0 indeterminate form.

(lim_x→0)(d(ln(1−2*x))/d(x)/d(x)/d(x))

(lim_x→0)((−2)/(1−2*x)/1)

5. Substitute x=0 into the simplified derivative expression.

(−2)/(1−2*(0))=−2

6. Exponentiate the result to find the limit of the original function, since (lim_)(ln(y))=L⇒(lim_)(y)=e^L

e^(−2)

Final Answer

(lim_x→0)(1−2*x)=e^(−2)

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