If the numerator and the denominator of a rational function have the same common factor, they can be factored. Let's take a look at the example.
lim_(x→ 1)x^2 - 1/x^2 + x - 2 = lim_(x→ 1)(x-1)(x+1)/(x-1)(x+2)
We can see that for x=1 both the numerator and the denominator are equal to 0. The most convenient method of finding such a limit is canceling the common factor (x-1).
lim _(x→ 1)(x-1)(x+1)/(x-1)(x+2) = lim_(x→ 1)x+1/x+2
Finally, we can substitute x=1 into the rational expression to find the limit.
1+1/1+2=2/3
This technique of dividing out the common factors is called the dividing out technique. Let's fill in the blank!
To find a limit of a rational function that has common factors in its numerator and denominator, use the dividing out technique.
