if the function f(x) defined by f(x)=((2^x)–1/x)+A,x<0. =log(2)+2B, x=0. =2((2^x)-1)((x^2)–x)/(x^2)((x^2)–1),x>0. is continuous at x=0,find the value of a and b
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Answer:
f(x)={
λ(x
2
−2x),ifx≤0
4x+1,ifx>0
If f is continuous at x=0, then
x→0
−
lim
f(x)=
x→0
+
lim
f(x)=f(0)
⇒
x→0
lim
λ(x
2
−2x)=
x→0
lim
(4x+1)=λ(0
2
−2x0)
⇒λ(0
2
−2⋅0)=4⋅0+1=0
⇒0=1=0, which is not possible
Therefore, there is no value of λ for which f is continuous at x=0
At x=1,
f(1)=4x+1=4⋅1+1=5
x→1
lim
(4x+1)=4⋅1+1=5
∴
x→1
lim
f(x)=f(1)
Therefore, for any values of λ, f is continuous at x=1
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