If lim _(x → 0) (ax - (e⁴ˣ - 1))/ax(e⁴ˣ - 1) exists and is equal to b, then the value of a - 2b is __________.

Solution

Maths sir · 4 steps

Expand e⁴ˣ-1 as a series and the x term in the numerator must die. That fixes a at once.

1. Expand the exponential. Use eᵗ = 1 + t + t²/2 + … with t = 4x. This gives e⁴ˣ - 1 = 4x + 8x² + …
2. Look at the orders. The numerator becomes (a - 4)x - 8x² - … and the denominator becomes ax(4x + 8x² + …) = 4a x² + … The denominator starts at x². If the numerator keeps an x term, the limit blows up.
3. Fix a. For the limit to exist the x term must vanish, so a - 4 = 0, giving a = 4.

   b = lim(x → 0) (-8x² - …)/(16x² + …) = -8/16 = -1/2

4. So. a - 2b = 4 - 2(-1/2) = 4 + 1 = 5.

The common slip: 3. Dropping the minus sign in front of 8x² makes b = 1/2, and then a - 2b comes out as 3, not 5.

The idea to keep. When a limit is given to exist, expand everything and match orders. Any numerator term of lower order than the denominator must have zero coefficient, and that fixes the unknown.

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