Compute lim_{x->0} (e^{x} - 1)/x.

Compute lim_{x->0} (e^{x} - 1)/x.

• A. 0
• B. ∞
• C. 1
• D. -1

Answer: (C)

When x is near 0, e^x is well approximated by its linear part: e^x ≈ 1 + x. So e^x - 1 is about x, and dividing by x gives something close to 1. As x approaches 0, this approximation becomes exact in the limit, giving the value 1. This can also be seen as the derivative of e^x at 0: lim (e^x - e^0)/(x - 0) = (d/dx e^x)|_{x=0} = e^0 = 1. A quick series expansion, e^x = 1 + x + x^2/2 + ..., leads to (e^x - 1)/x = 1 + x/2 + x^2/6 + ... → 1. The other possibilities don’t fit the behavior of e^x near zero.

When x is near 0, e^x is well approximated by its linear part: e^x ≈ 1 + x. So e^x - 1 is about x, and dividing by x gives something close to 1. As x approaches 0, this approximation becomes exact in the limit, giving the value 1. This can also be seen as the derivative of e^x at 0: lim (e^x - e^0)/(x - 0) = (d/dx e^x)|_{x=0} = e^0 = 1. A quick series expansion, e^x = 1 + x + x^2/2 + ..., leads to (e^x - 1)/x = 1 + x/2 + x^2/6 + ... → 1. The other possibilities don’t fit the behavior of e^x near zero.