Evaluate `lim_(x to 4) (x^(2) - 7x + 12)/(x^(2) - 16)`

Explanation

Here, the function is a rational function. Hence we first evaluate this function at the prescirbed point. If this is of the form `(0)/(0)` then we try to find the common factor in numerator and denominator and then rewrite the function by canelling the factors which are causing the limit to be of the form `(0)/(0)`.
We have `underset(x to 4)(lim) (x^(7) - 7x + 12)/(x^(2) - 16) = ((4)^(2) - 7 (4) + 12)/(16 -16) = (0)/(0)`
Which is of the form `(0)/(0)`
`:. underset(x to 4)(lim) (x^(2) - 7x + 12)/(x^(2) - 16) = underset(x to 4)(lim) ((x - 4)(x - 3))/((x + 4)(x - 4))` [Cancelling the non-zero factor (X - 4)]
`= underset(x to 4)(lim) (x - 3)/(x + 4) = (underset(x to 4)(lim)(x - 3))/(underset(x to 4)(lim)(x + 4)) = ((4 - 3))/((4 + 4)) = (1)/(8)`
`:. underset(x to 4)(lim) (x^(2) - 7x + 12)/(x^(2) - 16) = (1)/(8)`