Evaluate the Limit limit as x approaches 0 of (1+x)^(8/x)

Problem

(lim_x→0)(1+x)

Solution

1. Identify the form of the limit by substituting x=0 into the expression (1+x)^8/x which results in the indeterminate form 1^∞

2. Rewrite the expression using the natural logarithm and the exponential function to move the variable out of the exponent.

y=(1+x)^8/x

ln(y)=ln((1+x)^8/x)

ln(y)=8/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)((8*ln(1+x))/x)

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

(lim_x→0)((d(8)*ln(1+x))/d(x)/d(x)/d(x))

(lim_x→0)(8/(1+x)/1)

5. Substitute x=0 into the simplified derivative to find the limit of the logarithm.

8/(1+0)=8

6. Exponentiate the result to find the limit of the original expression, since ln((lim_)(y))=8 implies (lim_)(y)=e^8

Final Answer

(lim_x→0)(1+x)=e^8

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