Introducing your new favourite teacher - Teachoo Black, at only ₹83 per month
Example 7 - Chapter 5 Class 12 Continuity and Differentiability (Term 1)
Last updated at March 12, 2021 by Teachoo
Examples
Example 1
Example 2
Example 3
Example 4 Important
Example 5
Example 6
Example 7 You are here
Example 8
Example 9
Example 10
Example 11 Important
Example 12
Example 13 Important
Example 14
Example 15 Important
Example 16
Example 17 Important
Example 18
Example 19
Example 20 Important
Example 21
Example 22
Example 23 Important
Example 24
Example 25
Example 26 Important
Example 27
Example 28
Example 29 (i)
Example 29 (ii) Important
Example 29 (iii) Important
Example 29 (iv)
Example 30 Important
Example 31
Example 32 Important
Example 33 Important
Example 34
Example 35
Example 36 Important
Example 37 Important
Example 38
Example 39 Important
Example 40
Example 41 Important
Example 42 Important
Example 43
Example 44 (i)
Example 44 (ii) Important
Example 44 (iii) Important
Example 45 (i)
Example 45 (ii) Important
Example 45 (iii) Important
Example 46
Example 47 Important
Example 48
Transcript
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 > 0 Case 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 < 0 Case 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.
