Question

Suppose `f(x)={(a+bx, x<1), (4, x=1), (b-ax, x>1):}` and if `lim_(xrarr1) f(x)=f(1)`, what are the values of `a and b?`

Answer 1

Clearly, `f(1)=4.`
`lim_(xto1^(+))f(x)=lim_(hto0)f(1+h)=lim_(hto0){b-a(1+h)}=(b-a).`
`lim_(xto1^(-))f(x)=lim_(xto0)f(1-h)=lim_(hto0){a+b(1-h)}=(a+b).`
Now, `lim_(xto1)f(x)=f(1)=4`
`implieslim_(xto1^(+))f(x)=lim_(xto1^(-))f(x)=f(1)=4`
`implies(b-a)=(a+b)=4`
`impliesb-a=4andb+a=4impliesa=0,b=4.`
Hence, `a=0 and b=4.`