Example 17 - Discuss continuity of sine function - Class 12

Example 17 Discuss the continuity of sine function. Let 𝑓﷐𝑥﷯=﷐sin﷮𝑥﷯ Let c be any real number. We know that A function is continuous at 𝑥 = 𝑐 if L.H.L = R.H.L = 𝑓﷐𝑐﷯ i.e. ﷐lim﷮x→﷐𝑐﷮−﷯﷯ 𝑓﷐𝑥﷯= ﷐lim﷮x→﷐𝑐﷮+﷯﷯ 𝑓﷐𝑥﷯= 𝑓﷐𝑐﷯ Taking L.H.L ﷐lim﷮x→﷐𝑐﷮−﷯﷯𝑓﷐𝑥﷯ = ﷐lim﷮x→﷐𝑐﷮−﷯﷯ ﷐﷐sin﷮𝑥﷯﷯ Since ﷐sin﷮𝑥﷯ is defined for every real number Putting x = c − h = ﷐lim﷮h→0﷯ ﷐sin﷮﷐𝑐−ℎ﷯﷯ = ﷐lim﷮h→0﷯ ﷐﷐sin﷮𝑐﷯﷐cos﷮ℎ﷯−﷐cos﷮𝑐﷯﷐sin﷮ℎ﷯﷯ Putting ℎ=0 = ﷐sin﷮𝑐﷯﷐cos﷮0﷯−cos c . sin 0 = ﷐sin﷮𝑐﷯﷐1﷯−cos c . 0 = ﷐sin﷮𝑐﷯ Taking R.H.L ﷐lim﷮x→﷐𝑐﷮+﷯﷯𝑓﷐𝑥﷯ = ﷐lim﷮x→﷐𝑐﷮+﷯﷯ ﷐sin﷮𝑥﷯ Putting x = c + h = ﷐lim﷮h→0﷯﷐sin﷮﷐𝑐+ℎ﷯﷯ = ﷐lim﷮h→0﷯ (﷐sin﷮𝑐﷯﷐cos﷮ℎ﷯+﷐cos﷮𝑐﷯﷐sin﷮ℎ)﷯ Putting ℎ=0 = ﷐sin﷮𝑐﷯﷐cos﷮0﷯+cos c . sin 0 = ﷐sin﷮𝑐﷯﷐1﷯+cos c . 0 = ﷐sin﷮𝑐﷯ & 𝑓﷐𝑥﷯ = ﷐sin﷮𝑥﷯ 𝑓﷐𝑐﷯ = ﷐sin﷮𝑐﷯ Hence L.H.L = R.H.L = 𝑓﷐𝑐﷯ = ﷐lim﷮x→﷐𝑐﷮−﷯﷯ 𝑓﷐𝑥﷯=﷐lim﷮x→﷐𝑐﷮+﷯﷯ 𝑓﷐𝑥﷯= 𝑓﷐𝑐﷯ ⇒ 𝑓﷐𝑥﷯ is continuous for all real number So, 𝒔𝒊𝒏⁡𝒙 is continuous.