Question

How do you evaluate the limit `(x^2-7x+10)/(x-2)` as x approaches `2`?

Answer 1

`:. lim_(x rarr 2) (x^2-7x+10)/(x-2) = -3`

Explanation:

`lim_(x rarr 2) (x^2-7x+10)/(x-2) = lim_(x rarr 2) ((x-5)(x-2))/(x-2)`

When we evaluate the limit we look at the behaviour as `x` approaches `2` and we are not interested in what happens when `x=2` so we can cancel the `(x-2)` factor as `x!=2`

`:. lim_(x rarr 2) (x^2-7x+10)/(x-2) = lim_(x rarr 2) (x-5)`
`:. lim_(x rarr 2) (x^2-7x+10)/(x-2) = (2-5)`
`:. lim_(x rarr 2) (x^2-7x+10)/(x-2) = -3`

Answer 2

Factor the numerator to simplify the rational function

Explanation:

`x^2-7x+10 = (x-2)(x-5)`

`lim_(x->2) frac (x^2-7x+10) (x-2) = lim_(x->2) frac ((x-2)(x-5)) (x-2)`

`:. = lim_(x->2) (x-5) = -3`