| Step | Expression | Explanation |
|---|---|---|
| 1 | lim(x → 0) (eᵡ – 1 – x) / x² | Start with the original limit |
| 2 | 0 / 0 | Plug in x = 0 to check the form of the limit, which is indeterminate |
| 3 | Apply L’Hôpital’s Rule | Since the limit is indeterminate, we can apply L’Hôpital’s Rule |
| 4 | d/dx (eᵡ – 1 – x) = eᵡ – 0 – 1 | Differentiate the numerator to get eᵡ – 1 |
| 5 | d/dx (x²) = 2x | Differentiate the denominator to get 2x |
| 6 | lim(x → 0) (eᵡ – 1) / 2x | Substitute the derivatives into the limit |
| 7 | 0 / 0 | Plug in x = 0 again to check the form of the new limit, which is still indeterminate |
| 8 | Apply L’Hôpital’s Rule again | Since the new limit is still indeterminate, apply L’Hôpital’s Rule again |
| 9 | d/dx (eᵡ – 1) = eᵡ | Differentiate the new numerator to get eᵡ |
| 10 | d/dx (2x) = 2 | Differentiate the new denominator to get 2 |
| 11 | lim(x → 0) eᵡ / 2 | Substitute the new derivatives into the limit |
| 12 | e⁰ / 2 = 1 / 2 | Plug in x = 0 to find the limit is 1/2 |
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