lim (2-(x-3)^(1/2))/(x^2-49) as x->7- , please?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to substitute 7 for x in limit such that:

`lim_(x-gt7) (2-sqrt(x-3))/(x^2 - 49) = (2-sqrt(7-3))/(7^2-49)`

`lim_(x-gt7) (2-sqrt(x-3))/(x^2 - 49) = (2-2)/(49-49) = 0/0`

Notice that evaluating the limit yields an indetermination 0/0, hence, you may use l'Hospital's theorem such that:

lim_(x->7) (2-sqrt(x-3))/(x^2 - 49) = lim_(x->7) ((2-sqrt(x-3))')/((x^2 - 49)')

`lim_(x-gt7) ((2-sqrt(x-3))')/((x^2 - 49)') = lim_(x-gt7) (-1/(2sqrt(x-3)))/(2x)`

You need to substitute 7 for x in limit such that:

`lim_(x-gt7) (-1/(2sqrt(x-3)))/(2x) = (-1/(2sqrt(7-3)))/(2*7)`

`lim_(x-gt7) (-1/(2sqrt(x-3)))/(2x) = -1/56`

Hence, evaluating the limit of function using l'Hospital's theorem yields `lim_(x-gt7) (2-sqrt(x-3))/(x^2 - 49) = -1/56` .

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