Ace Your Limits with DAY 2002A Practice Test 2026 – Master Math Magic!

Near zero, the exponential behaves like its Taylor expansion: e^x = 1 + x + x^2/2 + higher-order terms. If you subtract 1 and x, the remaining leading piece is x^2/2 plus smaller terms. Dividing by x^2 gives 1/2 plus terms that go to zero as x → 0. Therefore the limit is 1/2.

If you prefer a calculus route, applying L'Hôpital twice also yields the same result: differentiating once gives (e^x − 1)/(2x), which is still 0/0 at 0; differentiating again gives e^x/2, and evaluating at x = 0 gives 1/2.