Let f: ℝ-{0} → ℝ be a function such that f(x)-6 f(1/x)=35/(3 x)-5/2. If the lim _(x → 0)(1/(α x)+f(x))=β ; α, β ∈ ℝ, then α+2 β is equal to

Solution

Maths sir · 4 steps

Swap x with 1/x to get a second equation, solve the pair for f(x), then kill the 1/x term in the limit.

1. Replace x by 1/x. The given equation holds for every nonzero x, so it also holds for 1/x. That gives f(1/x) - 6f(x) = 35x/3 - 5/2.
2. Eliminate f(1/x). From the second equation, f(1/x) = 6f(x) + 35x/3 - 5/2. Put this into the original equation to get -35f(x) - 70x + 15 = 35/3x - 5/2, so -35f(x) = 35/3x + 70x - 35/2.

   f(x) = -1/3x - 2x + 1/2

3. Make the limit finite. Now 1/(α x) + f(x) = (1/α - 1/3)1/x - 2x + 1/2. As x → 0 the 1/x term blows up unless its coefficient is zero, so α = 3. What remains tends to 1/2, so β = 1/2.
4. So. α + 2β = 3 + 2 × 1/2 = 4, which is option (d).

The idea to keep. When f(x) and f(1/x) appear together, replace x by 1/x and solve the two equations like a linear pair. For a finite limit, every term that blows up must cancel.

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