Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
NTA Abhyas JEE Main2020MathematicsContinuity and DifferentiabilityPractice

If f x = e 2 x + 2 x - 1 2 x + 2 x : x ≠ 0 1 : x = 0 , then (where, [.] represents the greatest integer function)

Options

  1. Al i m x → 0 + f x = - 1
  2. Bl i m x → 0 - f x = 1 e - 1
  3. Cf x is differentiable at x = 0
  4. Df x is discontinuous at x = 0

Correct answer

D. f x is discontinuous at x = 0

Step-by-step solution

l i m x → 0 + f x = l i m x → 0 + f 0 + h = l i m h → 0 + e 2 h + 2 h - 1 2 h + 2 h = l i m h → 0 + e 2 h - 1 2 h = 1 and l i m x → 0 - f x = l i m h → 0 + f 0 - h = l i m h → 0 e - 2 h - 2 h - 1 - 2 h - 2 h = l i m h → 0 e - 1 - 2 h - 1 - 1 - 2 h = e - 1 - 1 - 1 = 1 - 1 e Since, l i m x → 0 + f x ≠ l i m x → 0 - f x , therefore, f x is not continuous at x = 0 .

Practice Continuity and Differentiability on Quantrex Academy →

More from Continuity and Differentiability

Let R denote the set of all real numbers. Let f : R R be an arbitrary function and let g : R R be the function defined by g(x) = x f(x) , for all x R . Then which of the following 2026Consider the function f : (- 2 , 2 ) (- , ) defined by f(x) = (|x| + |x - 1|) x + [x x] , where [x x] is the greatest integer less than or equal to x x . Let be the total number of 2026For a real number , let [ ] denote the greatest integer less than or equal to . For a finite set S , let |S| denote the number of elements in the set S . Consider the functions f : 2026For the function f(x) = e^ |x| - |x| , x R , consider the following statements: Statement I: f is differentiable for all x R . Statement II: f is increasing in (- , - 2 ) . In the 2026Let f(x) = cases 1 3 , & x /2 b(1- x) ( -2x)^2 , & x > /2 cases . If f is continuous at x= /2 , then the value of ₀^ 3b-6 |x^2+2x-3| ,dx is: 2026Let f(x) = cases x^3 + 8 ; & x Then the number of points, where the function g f is discontinuous, is __________. 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 number of points where g is not continuous and is not differentiable are and respectively, then 2026The number of points, at which the function f(x) = 6x, 2 + 3x^2 + |x - 1| | |x^2 - 1 4 | | , x (- , ) , is not differentiable, is _____. 2026 Full Continuity and Differentiability list All NTA Abhyas JEE Main PYQs