Question 17: Find the relationship between a and b so that the function f defined by f(x) ={ax+1, if x≤ 3 bx+3,if x>3 is continuous at x = 3.
Class 12 - Math - Continuity and Differentiability
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At x = 3 L.H.L = lim x→3- (ax+1)
= lim h→0 a(3-h) +1 { putting x= 3-h
= lim h→0 3a-3h+1 = 3a+1
R.H.L = lim x→3+ f(x) = lim x→3+ (bx+3)
= lim h→0 [b(3+h)+3] { putting x = 3+h
= 3b+3
also f(3) = a(3) + 1 = 3a+1
Now f(x) is continuous at x=3
i.e., 3a+1 = 3b+3 => 3(a+b) = 2
=> a-b = 2/3 => a=b + 2/3
= lim h→0 a(3-h) +1 { putting x= 3-h
= lim h→0 3a-3h+1 = 3a+1
R.H.L = lim x→3+ f(x) = lim x→3+ (bx+3)
= lim h→0 [b(3+h)+3] { putting x = 3+h
= 3b+3
also f(3) = a(3) + 1 = 3a+1
Now f(x) is continuous at x=3
i.e., 3a+1 = 3b+3 => 3(a+b) = 2
=> a-b = 2/3 => a=b + 2/3
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