Question 18
If f (x) = x2 sin 1/π₯, where x β 0, then the value of the function f at x = 0, so that the function is continuous at x = 0, is
(A) 0 (B) β 1
(C) 1 (D) none of these
Given
f (x) = x2 sin 1/π₯, when x β 0
To find f (0)
f(x) is continuous at π₯=0
if L.H.L = R.H.L = π(0)
if limβ¬(xβ0^β ) π(π₯)=limβ¬(xβ0^+ ) " " π(π₯)= π(0)
LHL at x β 0
limβ¬(xβ0^β ) f(x) = limβ¬(hβ0) f(0 β h)
= limβ¬(hβ0) f(βh)
= limβ¬(hβ0) (ββ)^2 πππβ‘γπ/((βπ))γ
= limβ¬(hβ0) β^2 π
= 02 .π
= 0
RHL at x β 0
limβ¬(xβ0^+ ) f(x) = limβ¬(hβ0) f(0 + h)
= limβ¬(hβ0) f(h)
= limβ¬(hβ0) β^2 πππβ‘γπ/πγ
= limβ¬(hβ0) β^2 π
= 02. π
= 0
Since,
L.H.L = R.H.L = π(0)
0=π(0)
β΄ f (0) =π
So, the correct answer is (A)

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.