Evaluate the Limit limit as x approaches 0 of (1+x)^(1/x)

Problem

(lim_x→0)(1+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+x)^1/x

ln(y)=ln((1+x)^1/x)

ln(y)=1/x*ln(1+x)

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

(lim_x→0)(ln(y))=(lim_x→0)(ln(1+x)/x)

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

(lim_x→0)(d(ln(1+x))/d(x)/d(x)/d(x))

(lim_x→0)(1/(1+x)/1)

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

1/(1+0)=1

6. Exponentiate the result to find the limit of the original function, since ln(y)→1 implies y→e^1

(lim_x→0)(1+x)=e^1=e

Final Answer

(lim_x→0)(1+x)=e

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