Question 1
The function f (x) = {β 8(sinβ‘π₯/π₯ " + cos x, if x " β " 0" @π ", if x " =" 0" )β€ is continuous at x = 0, then the value of k is
(A) 3 (B) 2
(C) 1 (D) 1.5
At π = 0
f(x) is continuous at π₯=0
if L.H.L = R.H.L = π(0)
if if limβ¬(xβ0^β ) π(π₯) = limβ¬(xβ0^+ ) π(π₯) = π(0)
LHL at x β 0
(πππ)β¬(π±βπ^β ) f(x) = limβ¬(hβ0) f(0 β h)
= (πππ)β¬(π‘βπ) f(βh)
= limβ¬(hβ0) sinβ‘γ(ββ)γ/((ββ)) " + cos (β h)"
= (πππ)β¬(π‘βπ) γβπππγβ‘π/(βπ) " +" (πππ)β¬(π‘βπ) " cos h"
= limβ¬(hβ0) sinβ‘β/β " + " (πππ)β¬(ββ0) " cos h"
Using limβ¬(xβ0) sinβ‘π₯/π₯=1
= 1 + "cos 0"
= 1 + 1
= 2
RHL at x β 0
(πππ)β¬(πβπ^+ ) f(x) = limβ¬(hβ0) f(0 + h)
= (πππ)β¬(π‘βπ) f(h)
= limβ¬(hβ0) sinβ‘β/β " + cos h"
= (πππ)β¬(π‘βπ) πππβ‘π/π " + " (πππ)β¬(π‘βπ) " cos h"
Using limβ¬(xβ0) sinβ‘π₯/π₯=1
= 1 + "cos 0"
= 1 + 1
= 2
At π=π
π(π₯)=π
π(π)=π
Since f(x) is continuous at x = 0.
L.H.L = R.H.L = π(0)
π(0)=2
π=π
So, the correct answer is (B)

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.