limx→−1x2+6x+5x2−3x−4

To find the limit of the function as x approaches -1, we can substitute -1 into the function:

lim(x→-1) [(x^2 + 6x + 5) / (x^2 - 3x - 4)]

= [((-1)^2 + 6*(-1) + 5) / ((-1)^2 - 3*(-1) - 4)] = [(1 - 6 + 5) / (1 + 3 - 4)] = [0 / 0]

This is an indeterminate form, so we can't directly compute the limit. We need to simplify the function. We can do this by factoring the numerator and the denominator:

The numerator factors to (x + 1)(x + 5) and the denominator factors to (x + 1)(x - 4). So the function simplifies to:

(x + 1)(x + 5) / (x + 1)(x - 4)

We can cancel out the (x + 1) terms:

(x + 5) / (x - 4)

Now we can substitute -1 into the simplified function:

lim(x→-1) [(x + 5) / (x - 4)] = ((-1) + 5) / ((-1) - 4) = 4 / -5 = -4/5

So, lim(x→-1) [(x^2 + 6x + 5) / (x^2 - 3x - 4)] = -4/5.