Which of the following functions is not continuous at the origin?

  1. sin x
  2. x
  3. x sin x
  4. \(\frac{{\sin x}}{x}\)

Answer (Detailed Solution Below)

Option 4 : \(\frac{{\sin x}}{x}\)

Detailed Solution

Download Solution PDF

Concept:

  • A function is said to continuous at a given value if the function does not have a sudden change at that value.
  • If two functions are individually continuous then it’s summation difference, the product will be also continuous.
  • A function is continuous at x = a
    If L.H.L = R.H.L = f(a)
    i.e. \(\mathop {\lim }\limits_{x \to {a^ - }} f\left( x \right) = \mathop {\lim }\limits_{x \to {a^ + }} f\left( x \right) = f\left( a \right)\) 
  • If f(x) and g(x) are individually continuous then it’s f(x) ± g(x), f(x) ⋅ g(x) and \(\frac{{f\left( x \right)}}{{g\left( x \right)}}\;;g\left( x \right) \ne 0\) will be continuous.

Calculation:

Option (A):

Let f(x) = sin x

F1 Shubham.B 20-10-20 Savita D 1

From graph, at origin i.e. x = 0, sin x is continuous.

Option (B):

Let g(x) = x

F1 Shubham.B 20-10-20 Savita D 2

From graph at origin i.e. x = 0

g(x) = x is continuous.

Option (C):

Let h(x) = x sin x

h(x) = g(x) ⋅ f(x)

Since f(x) and g(x) is individually continuous at origin, so h(x) will be also continuous.

Option (D):

Let \(h\left( x \right) = \frac{{\sin x}}{x} = \frac{{f\left( x \right)}}{{g\left( x \right)}}\) 

Since sin x is continuous at origin and x is also continuous at origin and values are zero.

i.e. \(h\left( x \right) = \frac{{\sin x}}{x}\) 

At origin:

\(h\left( {x = 0} \right) = \frac{{\sin \left( 0 \right)}}{0} = \frac{0}{0}\)  is a indeterminate form)

Therefore, we can say that.

h(x) will be discontinuous at origin as at x = 0, h(x) become undefined.

So, option (D) is correct as \(\frac{{\sin x}}{x}\) is discontinuous at origin.

Latest BPSC Lecturer Updates

Last updated on May 9, 2025

-> The BPSC Lecturere DV will be held on 28th May 2025.

-> The written exam took place on 20th March 2025.

-> BPSC Lecturer notification was released for 6 vacancies.

-> The BPSC recruits eligible candidates for the post of Lecturer for various disciplines in Government Colleges across the state.

-> Candidates must attempt the BPSC Lecturer EC Mock Tests. Candidates can also download and practice from the BPSC Lecturer previous year papers

Get Free Access Now
Hot Links: teen patti royal - 3 patti teen patti gold download teen patti winner teen patti chart teen patti real cash