Question

How do you evaluate `(x^2-14x-32) / (xsqrt(x -64)` as x approaches 16?

Answer 1

Note that as `xrarr16`, the expression takes on imaginary values. (The denominator involves `sqrt(x-64)`.)

Explanation:

In a course in the calculus or real numbers, this might be a typographic error or the desired answer might be "is not defined".

Many introductory courses give definitions of limits only for real valued functions of real variables. In such a course, the domain of the expression is `(64, oo)`, so the limit at `16` is not defined.

In a course that includes imaginary complex numbers, the fact that the numerator goes to `0` while the denominator does not should be enough to conclude that the limit is `0`. (My complex analysis is weak, but I am sure that this limit is `0`, but I'd have to work to get the proof.)

Note that the numerator can be factored:

`x^2-14-32 = (x-16)(x+2)`

at which point it is easier to see that it goes to `0` as `xrarr16`.