Let `f(x){:{(2x+3",",xle0),(3(x+1)",",xgt0.):}` Find `(i) lim_(xrarr0)f(x)" "(ii)lim_(xrarr1)f(x)`

Let `f(x){:{(2x+3",",xle0),(3(x+1)",",xgt0.):}` Find `(i) lim_(xrarr0)

1.	Let `f(x){:{(2x+3",",xle0),(3(x+1)",",xgt0.):}` Find `(i) lim_(xrarr0)f(x)" "(ii)lim_(xrarr1)f(x)`
Answer» (I) We have `lim_(xto0^(+))f(x)=lim_(hto0)3(0+h+1)=lim_(hto0)3(h+1)=3.` `lim_(xto0^(-))f(x)=lim_(hto0)3(0-h+1)=lim_(hto0)3(-h+1)=3.` `thereforelim_(xto0^(+))(x)=lim_(xto0^(-))f(x)=3.` Hence, `lim_(xto0)f(x)=3.` (ii) We have `limf(x)lim_(hto0)f(1+h)=lim_(hto0)3(1+h+1)=lim_(hto0)3(2+h)=6.` `lim_(xto1^(-))f(x)=lim_(hto0)f(1-h)=lim_(hto0)3(1-h+1)=lim_(hto0)3(2-h)=6.` `thereforelim_(xto1^(+))f(x)=lim_(xto1)f(x)=6.` Hence `lim_(xto1)f(x)=6.`