Evaluate the Limit limit as x approaches 0 of (2^x-1)/x

Problem

(lim_x→0)((2−1)/x)

Solution

1. Identify the form of the limit by substituting x=0 into the expression.

2. Evaluate the numerator and denominator at x=0 which results in (2−1)/0=(1−1)/0=0/0

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

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

5. Recall the derivative formula d(ax)/d(x)=ax*ln(a)

6. Calculate the derivative of the numerator: (d(2)−1)/d(x)=2*ln(2)

7. Calculate the derivative of the denominator: d(x)/d(x)=1

8. Substitute these derivatives back into the limit expression.

9. Evaluate the new limit as x approaches 0

(lim_x→0)((2*ln(2))/1)

2*ln(2)=1⋅ln(2)

Final Answer

(lim_x→0)((2−1)/x)=ln(2)

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