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Evaluate: (without using Hopital's rule)

d) lim (6-3x)/(((5x+6)^1/2)-4)
x->2

1 answer

(6-3x)/[sqrt(5x+6)-4] =

(6-3x)[sqrt(5x+6)+4]/{[sqrt(5x+6)+4][sqrt(5x+6)-4]} =

(6-3x)[sqrt(5x+6)+4]/(5x+6 - 16) =

(6-3x)[sqrt(5x+6)+4]/(5x-10)

The limit of x to 2 of

(6-3x)/(5x-10)

can be computed by expressing e.g. the denominator in terms of the numerator:

5 x - 10 = -5/3 (6-3x)

So, that limit is -3/5 and multiplying with the value of sqrt(5x+6)+4 at x = 2 gives

-5/3 * 8 = -40/3

for the limit of the original function.
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