If lim_x → 1 (x^4 - 1)/(x - 1) = lim_x → k (x^3 - k^3)/(x^2 - k^2), then k is

If lim_x → 1 (x^4 - 1)/(x - 1) = lim_x → k (x^3 - k^3)/(x^2… — JEE Main 2019 Mathematics PYQ

JEE Main 2019 · 10 April · Shift I

\lim_{x \to 1} \frac{x^4 - 1}{x - 1} = \lim_{x \to k} \frac{x^3 - k^3}{x^2 - k^2}

, then

k

A.
$\frac{4}{3}$
B.
$\frac{3}{8}$
C.
$\frac{3}{2}$
D.
$\frac{8}{3}$

Show answer and solution

Correct option: D

Solution

The left limit uses the standard formula: lim(x→1) (x⁴-1)/(x-1) = 4·1³ = 4.

For the right side, apply lim(x→k) (xⁿ-kⁿ)/(x-k) = n·kⁿ⁻¹ to numerator and denominator separately:

lim(x→k) (x³-k³)/(x²-k²) = [3k²]/[2k] = 3k/2.

Setting 3k/2 = 4 gives k = 8/3.

[JEE Main 2019 · 10 April · Shift I]

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