Evaluate the limit, if it exists.

Question

lim x->16 (4-sqrt(x))/(16x-x^2)

Answer 1

To evaluate the limit, we substitute 16 in place of x in the expression and simplify:

lim x->16 (4 - sqrt(x))/(16x - x^2) = (4 - sqrt(16))/(16(16) - 16^2)
= (4 - 4)/(256 - 256)
= 0/0

The resulting expression is of the form 0/0, which means that we can use L'Hospital's Rule to determine the limit. Taking the derivative of both the numerator and denominator:

lim x->16 (4 - sqrt(x))/(16x - x^2) = lim x->16 (-1/2sqrt(x))/(16 - 2x) / (16 - 2x)
= lim x->16 -1/[2sqrt(x)(16 - 2x)]

Now we substitute 16 back into the expression:

lim x->16 -1/[2sqrt(x)(16 - 2x)] = -1/[2sqrt(16)(16 - 2(16))]
= -1/[2(4)(16 - 32)]
= -1/[-64]
= 1/64

Therefore, the limit of the expression as x approaches 16 is 1/64.