Let f(x) = begincases x2-1, 0

Question

Let f(x) = begincases x2-1, 0 < x < 2 [2ex] 2x+3, 2 le x < 3 endcases, the quadratic equation whose roots are displaystyle limx → 2-f(x) and displaystyle limx → 2+f(x) is (A) x2 -6x +9=0 (B) x2 -7x +8=0 (C) x2 -14x +49=0 (D) x2 -10x +21=0

Tardigrade Admin · Accepted Answer

displaystyle limx → 2-f(x) = displaystyle limh → 0(2-h)2-1 = displaystyle limh → 04+h2-4h-1=3 displaystyle limx → 2+f(x)= displaystyle limh → 02(2+h)+3 = displaystyle limh → 04+2h+3=7 ∴ Required quadratic equation is x2 - ( 3 + 7)x + ( 3 × 7) = 0 ⇒ x2 -10x+21=0

Q. Let $f (x) = ⎩ ⎨ ⎧ x^{2} - 1, 2 x + 3, 0 < x < 2 2 \leq x < 3$ , the quadratic equation whose roots are $x \to 2^{-} lim f (x)$ and $x \to 2^{+} lim f (x)$ is

$x^{2} - 6 x + 9 = 0$

$x^{2} - 7 x + 8 = 0$

$x^{2} - 14 x + 49 = 0$

$x^{2} - 10 x + 21 = 0$

$x \to 2^{-} lim f (x)$ $= h \to 0 lim (2 - h)^{2} - 1$
$= h \to 0 lim 4 + h^{2} - 4 h - 1 = 3$
$x \to 2^{+} lim f (x)$ $= h \to 0 lim 2 (2 + h) + 3$
$= h \to 0 lim 4 + 2 h + 3 = 7$
$∴$ Required quadratic equation is
$x^{2} - (3 + 7) x + (3 \times 7) = 0$
$\Rightarrow x^{2} - 10 x + 21 = 0$

Q. Let f(x)=⎩⎨⎧​x2−1,2x+3,​0<x<22≤x<3​, the quadratic equation whose roots are x→2−lim​f(x) and x→2+lim​f(x) is

x2−6x+9=0

x2−7x+8=0

x2−14x+49=0

x2−10x+21=0