1. Chapter 5 Class 12 Continuity and Differentiability
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Ex 5.1, 18 For what value of λ is the function defined by 𝑓 𝑥﷯= λ 𝑥﷮2﷯−2𝑥﷯, 𝑖𝑓 𝑥≤0﷮&4𝑥−1, 𝑖𝑓 𝑥>0﷯﷯ continuous at x = 0? What about continuity at x = 1? We have, 𝑓 𝑥﷯= λ 𝑥﷮2﷯−2𝑥﷯, 𝑖𝑓 𝑥≤0﷮&4𝑥−1, 𝑖𝑓 𝑥>0﷯﷯ (i) At 𝒙=𝟎 f is continuous at 𝑥=0 if L.H.L = R.H.L = 𝑓 0﷯ i.e. if lim﷮x→ 0﷮−﷯﷯ 𝑓 𝑥﷯ = lim﷮x→ 0﷮+﷯﷯ 𝑓 𝑥﷯ = 𝑓 0﷯ Thus, L.H.L ≠ R.H.L ∴ f is not continuous at x = 0. So, for any value of λ∈𝐑 , f is discontinuous at x = 0. (ii) At 𝒙=𝟏 𝑓 𝑥﷯ = 4𝑥+1 f is continuous at 𝑥=1 if lim﷮x→1﷯ 𝑓 𝑥﷯ = 𝑓 1﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.