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

Problem

(lim_x→0)((ex−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 (e0−1)/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 separately.

5. Calculate the derivative of the numerator (d(ex)−1)/d(x)=ex and the derivative of the denominator d(x)/d(x)=1

6. Substitute these derivatives back into the limit to get (lim_x→0)((ex)/1)

7. Evaluate the new limit by substituting x=0 into the simplified expression.

8. Simplify the result to find e0=1

Final Answer

(lim_x→0)((ex−1)/x)=1

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