Determine whether each function Is continuous at the given j

Determine whether each function Is continuous at the given jr-value{s). Justify using the continuity test If discontinuous, identity the type of discontinuity as infinite, jump, or removable. f(x) = x^2 - 3x; x = 4.

Solution

29)f(x)=x/(x+7)

lim_x->0^-f(x)

=lim_x->0^-x/(x+7)

=0/(0+7)

=0

lim_x->0⁺f(x)

=lim_x->0⁺x/(x+7)

=0/(0+7)

=0

f(0)=0/(0+7)

=0

lim_x->0^-x/(x+7) =lim_x->0⁺x/(x+7)=f(0)

function is continous at x =0

lim_x->7^-f(x)

=lim_x->7^-x/(x+7)

=7/(7+7)

=1/2

lim_x->7⁺f(x)

=lim_x->7⁺x/(x+7)

=7/(7+7)

=1/2

f(7)=7/(7+7)

=1/2

lim_x->7^-x/(x+7) =lim_x->7⁺x/(x+7)=f(7)

function is continous at x =7

30)f(x)=x/(x²-4)

lim_x->2^-f(x)

=lim_x->2^- x/(x²-4)

=2/(0^-)

=-infinity

lim_x->2⁺f(x)

=lim_x->2⁺ x/(x²-4)

=2/(0⁺)

=infinity

f(2) doesnot exist

lim_x->2^-f(x) not equal to lim_x->2f(x)

so f(x) is discontinous at x =2

lim_x->4^-f(x)

=lim_x->4^- x/(x²-4)

=4/(16-4)

=4/12

=1/3

lim_x->4⁺f(x)

=lim_x->4⁺ x/(x²-4)

=4/(16-4)

=4/12

=1/3

f(4)=1/3

lim_x->4^-f(x) =lim_x->4⁺f(x)=f(4)

function is continous at x =4