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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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