A piecewise function is given. Use properties of limits to find the shown indicated limit, or

nitraiddQ 2021-02-03 Answered

A piecewise function is given. Use properties of limits to find the shown indicated limit, or state that the limit does not exist. f(x)={x2+73ifx<14xifx1
limx1f(x)

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Expert Answer

Maciej Morrow
Answered 2021-02-04 Author has 98 answers

Given
f(x)={x2+73ifx<14xifx1
we know for any given function g(x) limit at given point x=h exist only
if, limx>hg(x)=limx>h+g(x),
thus, we are finding left hand limit and right hand limit one by one at x=1, if both are coming same then limit will exist other limit of given function will not exist
so,
L.H.L=limx1f(x)
(if x<1,thenf(x)=x2+73
=limx1x2+73
=12+73
=1+73
=83
=2
Now finding right hand limit also
R.H.L=limx1+f(x)
(if x>1 then f(x)=4x)
=limx1+(4x)
=limx+4(1)
=4
here
L.H.L != R.H.L.
hence, for given function limit at x=1 doesn't exists.

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