Question

How do you find the limit of `(x^2-5x+4)/(x^2-2x-8)` as `x->4`?

Answer 1

Factor, simplify and try again.

Explanation:

When we try to evaluate by substitution, we get

`lim_(xrarr4)(x^2-5x+4) = 0` and

`lim_(xrarr4)(x^2-2x-8) = 0` .

The initial form of `lim_(xrarr4)(x^2-5x+4) /(x^2-2x-8)` is the indeterminate form `0/0`.

Because `4` is a zero of both polynomials (the numerator and the denominator), we can be sure that `x-4` is a factor of both. Therefore we can simplify the ration and try again to evaluate the limit.

`lim_(xrarr4)(x^2-5x+4) /(x^2-2x-8) = lim_(xrarr4)((x-4)(x-1)) /((x-4)(x+2))`

`= lim_(xrarr4)(x-1) /(x+2) = (4-1)/(4-2) = 3/2`