Example 7 - Chapter 5 Class 12 Continuity and Differentiability
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 7 Is the function defined by f (x) = |x|, a continuous function? f(x) = |𝑥| = {█(−𝑥, 𝑥<0@𝑥, 𝑥≥0)┤ Since we need to find continuity at of the function We check continuity for different values of x When x = 0 When x < 0 When x > 0Case 1 : When x = 0 f(x) is continuous at 𝑥 =0 if L.H.L = R.H.L = 𝑓(0) if lim┬(x→0^− ) 𝑓(𝑥)=lim┬(x→0^+ ) " " 𝑓(𝑥)= 𝑓(0) Since there are two different functions on the left & right of 5, we take LHL & RHL . LHL at x → 0 lim┬(x→0^− ) f(x) = lim┬(h→0) f(0 − h) = lim┬(h→0) f(−h) = lim┬(h→0) |−ℎ| = lim┬(h→0) ℎ = 0 RHL at x → 0 lim┬(x→0^+ ) f(x) = lim┬(h→0) f(0 + h) = lim┬(h→0) f(h) = lim┬(h→0) |ℎ| = lim┬(h→0) ℎ = 0 & 𝑓(0) = 0 Hence, L.H.L = R.H.L = 𝑓(0) ∴ f is continuous at x=0 Case 2 : When x < 0 For x < 0, f(x) = −x Since this a polynomial It is continuous ∴ f(x) is continuous for x < 0Case 3 : When x > 0 For x > 0, f(x) = x Since this a polynomial It is continuous ∴ f(x) is continuous for x > 0 Hence, 𝑓(𝑥)= |𝑥| is continuous at all points. i.e. f is continuous at 𝒙 ∈ R.
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About the Author
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 and Computer Science at Teachoo