Calculus 1 help, lim as x approaches 4 of (1/((sqrtx)-2)) - (4/(x-4))?

Question

i know the answer is 1/4 but i don't know how to get it. help if you can please.

Answer 1

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lim(x-->4) [1/(sqrt(x) - 2) - 4/(x - 4)]
= lim(x-->4) [(sqrt(x) + 2)/(x - 4) - 4/(x - 4)]
= lim(x-->4) [(sqrt(x) - 2)/(x - 4)]
= lim(x-->4) (sqrt(x) - 2)/[(sqrt(x) + 2)(sqrt(x) - 2)]
= lim(x-->4) 1/(sqrt(x) + 2)
= 1/(sqrt(4) + 2)
= 1/4.

I hope that helps!

Answer 2

1/(√x -2)
= 1/(√x -2) * (√x +2)/(√x +2)
= 1(√x +2) / [(√x -2)(√x +2)]
= (√x +2) / (x - 4)

1/(√x -2) - 4/(x-4)
= (√x +2) / (x - 4) - 4/(x-4)
= (√x +2 -4) / (x - 4)
= (√x -2) / (x - 4)
= (√x -2)(√x +2) / [(x - 4)(√x +2)]
= (x - 4) / [(x - 4)(√x +2)]
= 1/(√x +2)

Therefore
lim x→4 1/(√x -2) - 4/(x-4)
= lim x→4 1/(√x +2)
= 1/(√4+2) = 1/(2+2) = 1/4