lim _(x → ∞) (2 x²-3 x+5)(3 x-1)^(x/2)/((3 x²+5 x+4) √(3 x+2)ˣ) is equal to :

Solution

Maths sir · 4 steps

Split it into a polynomial ratio and a 1^∞ power, then handle each piece separately.

1. Rewrite the root. √(3x+2)ˣ = (3x+2)^(x/2), which has the same power as (3x-1)^(x/2). So the limit is a product of two parts: (2x²-3x+5)/(3x²+5x+4) and ((3x-1)/(3x+2))^(x/2).
2. Polynomial part. The top and bottom both have degree 2, so the limit is the ratio of the leading coefficients, which is 2/3.
3. Power part. The base tends to 1 and the power tends to ∞, so this is the 1^∞ form. Use lim fᵍ = eˡⁱᵐᵍ⁽ᶠ⁻¹⁾, with f - 1 = (3x-1)/(3x+2) - 1 = -3/(3x+2).

   lim(x → ∞) x/2 · -3/(3x+2) = lim(x → ∞) -3x/(6x+4) = -1/2

4. So. The power part tends to e^(-1/2). The limit is 2/3 × e^(-1/2) = 2/3√e, which is option (b).

The idea to keep. When the base tends to 1 and the power tends to infinity, use eˡⁱᵐᵍ⁽ᶠ⁻¹⁾. Before that, write any root as a power so that matching powers can be combined into a single base.

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