Question 1
The function f (x) = {■8(sin𝑥/𝑥 " + cos x, if x " ≠" 0" @𝑘 ", if x " =" 0" )┤ is continuous at x = 0, then the value of k is
(A) 3 (B) 2
(C) 1 (D) 1.5
At 𝒙 = 0
f(x) is continuous at 𝑥=0
if L.H.L = R.H.L = 𝑓(0)
if if lim┬(x→0^− ) 𝑓(𝑥) = lim┬(x→0^+ ) 𝑓(𝑥) = 𝑓(0)
LHL at x → 0
(𝒍𝒊𝒎)┬(𝐱→𝟎^− ) f(x) = lim┬(h→0) f(0 − h)
= (𝒍𝒊𝒎)┬(𝐡→𝟎) f(−h)
= lim┬(h→0) sin〖(−ℎ)〗/((−ℎ)) " + cos (− h)"
= (𝒍𝒊𝒎)┬(𝐡→𝟎) 〖−𝒔𝒊𝒏〗𝒉/(−𝒉) " +" (𝒍𝒊𝒎)┬(𝐡→𝟎) " cos h"
= lim┬(h→0) sinℎ/ℎ " + " (𝑙𝑖𝑚)┬(ℎ→0) " cos h"
Using lim┬(x→0) sin𝑥/𝑥=1
= 1 + "cos 0"
= 1 + 1
= 2
RHL at x → 0
(𝒍𝒊𝒎)┬(𝒙→𝟎^+ ) f(x) = lim┬(h→0) f(0 + h)
= (𝒍𝒊𝒎)┬(𝐡→𝟎) f(h)
= lim┬(h→0) sinℎ/ℎ " + cos h"
= (𝒍𝒊𝒎)┬(𝐡→𝟎) 𝒔𝒊𝒏𝒉/𝒉 " + " (𝒍𝒊𝒎)┬(𝐡→𝟎) " cos h"
Using lim┬(x→0) sin𝑥/𝑥=1
= 1 + "cos 0"
= 1 + 1
= 2
At 𝒙=𝟎
𝑓(𝑥)=𝑘
𝒇(𝟎)=𝒌
Since f(x) is continuous at x = 0.
L.H.L = R.H.L = 𝑓(0)
𝑓(0)=2
𝒌=𝟐
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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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