lim x → 0 (e-(1+2 x)^(1/(2 x)))/x is equal to

Solution

Maths sir · 3 steps

The bracket tends to e, so expand it to first order in x and see what survives after dividing by x.

1. Take the log. Let y = (1+2x)^(1/2x). Then ln y = ln(1+2x)/2x. Use ln(1+t) = t - t²/2 + … with t = 2x, which gives ln y = 1 - x + ….
2. Back to y. So y = e^(1 - x + …) = e · e^(-x + …). Since e⁻ˣ ≈ 1 - x for small x, this gives y ≈ e(1 - x) to first order.

   (e - (1+2x)^(1/2x))/x ≈ (e - e(1-x))/x = ex/x = e

3. So. The terms we dropped are of order x², so they vanish after dividing by x. The limit equals e, which is option (d).

The idea to keep. When (1+t)^(1/t) sits next to e in a 0/0 form, use (1+t)^(1/t) ≈ e(1 - t/2). Keep the first-order term, because it is the whole answer.

Answer from the NTA final answer key. Solution by PrepWiser.