evaluate the limit

Question

lim x-->16 (4- √x)/(16x-x^2

Answer 1

Multiply the numerator and denominator by 4+√x, the conjugate of the numerator.

 

Simplifying, we get (16-x)/[x(16-x)(4+√x)] = 1/[x(4+√x)]

 

So, the given limit is equivalent to lim_x→16[1/(x(4+√x))] = 1/128.

Answer 2

Since this is an indeterminate form, you can use L'Hopital's rule.  Are you familiar with it?

 

lim_x→16 [(4 - √x) / (16x - x²)] = lim_x→16 [(-1/2)x^-1/2 / (16 - 2x)] = (-1/2) / [√16 (16 - 32)] = 1/128