1.

Find `lim_(xrarr0) f(x)` and `underset(xrarr1)f(x)`. `f(x)={{:(2x+3,xle0),(3(x+1),xgt0):}`

Answer» `f(x)={{:(2x+3,xle0),(3(x+1),xgt0):}`
at x=0
LHL`=underset(xrarr0^(-))"lim"f(x)`
`=underset(hrarr0)"lim"f(0-h)`
Let `0-h=x `
`rArr0-hrarr0`
`rArr hrarr0`
`underset(hrarr0)"lim"2(0-h)+3=2(0)+3=3`
RHL`=underset(xrarr0^(+))f(x)`
`=underset(hrarr0)"lim"f(0+h)`
`=underset(hrarr0)"lim"3(0+h+1)`
`=3(0+1)=3`
`because LHL=RHL=3`
` therefore underset(xrarr0)"lim"f(x)=3`
at x=1
LHL`=underset(xrarr1^(1-))"lim"f(x)`
`=underset(hrarr0)"lim"f(1-h)`
`=underset(hrarr0)"lim"3(1-h+1)`
`=3(2-0) =6`
RHL`=underset(xrarr1^(+))"lim"f(x)`
`=underset(hrarr0)"lim"f(1+h)`
`=underset(hrarr0)"lim"3(1+h+1)`
`=3(2+0)=6`
`because LHL=RHL=6`
`therefore underset(xrarr1)"lim"f(x)=6`
Let `1+h=x`
`rArr1+hrarr1`
`rArr hrarr0`


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