Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Continuity and Differentiability

NTA Abhyas 2020 · Mathematics · Continuity and Differentiability

The function f(x) = _ n → ∞ (x - 2) ^ 2n - 1 (x - 2) ^ 2n + 1 ( n ∈...

The function f\left(x\right) = \lim_{n \to \infty}\dfrac{{ \left(x - 2\right)}^{2n} - 1}{{ \left(x - 2\right)}^{2n} + 1} \left(\forall n \in \mathbb{N}\right) is discontinuous at&

Options

  1. Ax=1x = 1 only
  2. Bx=3x = 3 only
  3. Cx=1x = 1 and 33
  4. Dx=0,1x = 0,1 and 22

Correct answer

C. x=1x = 1 and 33

Step-by-step solution

Given, f(x)=lim⁡n→∞[(x−2)2]n−1[(x−2)2]n+1f\left(x\right) = \lim_{n \to \infty}\dfrac{{\left[{\left(x - 2\right)}^{2}\right]}^{n} - 1}{{\left[{\left(x - 2\right)}^{2}\right]}^{n} + 1}
⇒f(x)=lim⁡n→∞1−1[(x−2)2]n1+1[(x−2)2]n⇒ f\left(x\right) = \lim_{n \to \infty}\dfrac{1 - \dfrac{1}{\left[\left(x - 2\right)^{2}\right]n}}{1 + \dfrac{1}{\left[\left(x - 2\right)^{2}\right]n}}
⇒f(x)=⇒ f\left(x\right) =\left\{−1,0,1,\begin{array}{ll}- 1, \\ 0, \\ 1,\end{array}0≤(x−2)21\begin{array}{ll}0 \le \left(x - 2\right)^{2} 1\end{array}\right)
⇒ f\left(x\right) = \left\{\begin{cases}1, & x 3\end{cases}\right)<br />Thus,f \left(x\right)isdiscontinuousatis discontinuous atx = 1,3
Practice Continuity and Differentiability on Quantrex Academy →

More from Continuity and Differentiability

Let ℝ denote the set of all real numbers. Let f : ℝ arrow ℝ be an arbitrary function and… JEE Advanced 2026Consider the function f : (-(π)/(2), (π)/(2)) arrow (-∞, ∞) defined by f(x) = (|x| + |x… JEE Advanced 2026For a real number α , let [α] denote the greatest integer less than or equal to α . For a… JEE Advanced 2026For the function f(x) = e^ |x| - |x| , x ∈ ℝ , consider the following statements… JEE Main 2026Let f(x) = cases (1)/(3), & x ≤ π/2 (b(1- x))/((π-2x)^2), & x > π/2 cases . If f is… JEE Main 2026Let f(x) = cases x^3 + 8 ; & x Then the number of points, where the function g ∘ f is… JEE Main 2026Let f(x)= cases e^ x-1 , & x<0 x^2-5x+6, & x ≥ 0 cases and g(x)=f(|x|)+|f(x)| . If the… JEE Main 2026The number of points, at which the function f(x) = \ 6x, 2 + 3x^2\ + |x - 1|| |x^2… JEE Main 2026The number of points in the interval [2, 4] , at which the function f(x) = [x^2 - x… JEE Main 2026Let f(x)= _ θ arrow 0 ( π x-x^ ((2)/(θ)) (x-1) 1+x^ ((2)/(θ)) (x-1) ), x ∈ R . Consider… JEE Main 2026Let [t] denote the greatest integer less than or equal to t . If the function f(x)=\… JEE Main 2026If the function f(x)= e^ x (e^ x-x -1)+ _ e ( x+ x)-x x-x is continuous at x=0 , then the… JEE Main 2026Let α, β ∈ ℝ be such that the function f(x)= cases 2 α(x^ 2 -2)+2 β x &, x be… JEE Main 2026If f(x)=\ array cc a|x|+x^ 2 -2( |x|)( |x|) x &, x ≠ 0 b &, x=0 array . is continuous at… JEE Main 2026Let f(x)= cases a x^ 2 +2 a x+3 4 x^ 2 +4 x-3 &, x ≠-(3)/(2), (1)/(2) ~b &, x=-(3)/(2)… JEE Main 2026Let [·] denote the greatest integer function, and let f(x)= \ 2 x, x^ 2 \ . Let S = \ x… JEE Main 2026 Full Continuity and Differentiability list Continuity and Differentiability JEE Maths PYQsJEE Mathematics PYQs All NTA Abhyas JEE Main PYQs