lim[(2x2 + 3x + 1)/(x2 - 2x - 3)] (as x → -1) is of the form 0/0 if -1 is substituted for x. ( 0/0 is an indeterminate form )
When a limit has the form 0/0, we can try to simplify the function that we are taking the limit of so that we no longer have 0/0 when we plug in the given value of x.
In this particular problem, since the numerator and denominator are both equal to zero when x = -1, x - (-1) is a factor of both the numerator and denominator which will cancel out Then, substituting -1 for x into the resulting expression will give us the value of the limit.
lim [(2x+1)(x+1)/((x-3)(x+1))] (as x → -1)
= lim[(2x+1)/(x-3)] (as x → -1)
= [2(-1)+1]/(-1-3) = -1/(-4) = 1/4
