Ex 5.1, 24 - Chapter 5 Class 12 Continuity and Differentiability

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Ex 5.1, 24 Determine if f defined by 𝑓 𝑥﷯= 𝑥2 sin﷮ 1﷮𝑥﷯﷯, 𝑖𝑓 𝑥≠0﷮&0, 𝑖𝑓 𝑥=0﷯﷯ is a continuous function? 𝑓 𝑥﷯= 𝑥2 sin﷮ 1﷮𝑥﷯﷯, 𝑖𝑓 𝑥≠0﷮&0, 𝑖𝑓 𝑥=0﷯﷯ At x = 0 A function is continuous at x = 0 if L.H.L = R.H.L = 𝑓 0﷯ i.e. lim﷮x→ 0﷮−﷯﷯ 𝑓 𝑥﷯= lim﷮x→ 0﷮+﷯﷯ 𝑓 𝑥﷯= 𝑓 0﷯ L.H.L lim﷮x→ 0﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 0﷮−﷯﷯ 𝑥2 sin﷮ 1﷮𝑥﷯﷯ Putting 𝑥 = 0 − ℎ = lim﷮ℎ→0﷯ 0−h﷯﷮2﷯ . sin﷮ 1﷮0 − ℎ﷯﷯﷯ = lim﷮ℎ→0﷯ −h﷯﷮2﷯ . sin﷮ 1﷮− ℎ﷯﷯﷯ = lim﷮ℎ→0﷯ ℎ﷮2﷯ . sin﷮ 1﷮− ℎ﷯﷯﷯ = lim﷮ℎ→0﷯ − ℎ﷮2﷯ . sin . 1﷮ℎ﷯﷯ = lim﷮ℎ→0﷯ − ℎ﷮2﷯ . 𝑘 Putting h = 0 = 0﷮2﷯. 𝑘 = 0 ∴ LHL = 0 R.H.L lim﷮x→ 0﷮+﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 0﷮+﷯﷯ 𝑥2 sin﷮ 1﷮𝑥﷯﷯ Putting 𝑥 = 0+ℎ = lim﷮ℎ→0﷯ 0+h﷯﷮2﷯ . sin﷮ 1﷮0 + ℎ﷯﷯﷯ = lim﷮ℎ→0﷯ ℎ﷮2﷯ . sin ﷮ 1﷮ℎ﷯﷯ = lim﷮ℎ→0﷯ ℎ﷮2﷯ . 𝑘 Putting h = 0 = 0﷮2﷯. 𝑘 = 0 & 𝑓 0﷯ = 0 Thus L.H.L = R.H.L = 𝑓 0﷯ Hence 𝒇 𝒙﷯ is continuous for all real value of 𝒙 .