Calculus: Early Transcendentals, Chapter 2, 2.3, Section 2.3, Problem 14

lim_(x->-1) (x^2 - 4x)/(x^2 - 3x - 4)
sol:
lim_(x->-1) (x^2 - 4x)/(x^2 - 3x - 4)
=>lim_(x->-1) (x(x-4) )/(x^2 - 4x+x - 4)
=> lim_(x->-1)((x(x-4))/((x-4)(x+1)))
=>lim_(x->-1)((x)/(x+1))
In order for the limit to exist , the condition is
lim_(x->-1^-)((x)/(x+1)) = lim_(x->-1^+)((x)/(x+1))
let us see whether this condition is satisfied or not.
so, taking
lim_(x->-1^-)((x)/(x+1))
as x-> -1^- , in the denominator (x+1) will be a negative quantity approaching to 0
so, lim_(x->-1^-)((x)/(x+1)) = (-1)/(-1+1) = -oo
similarly,
lim_(x->-1^+)((x)/(x+1))
as x-> -1^+ , in the denominator (x+1) will be a positive quantity approaching to 0
so, lim_(x->-1^+)((x)/(x+1)) = (-1)/(-1+1)= +oo
as lim_(x->-1^-)((x)/(x+1)) != lim_(x->-1^+)((x)/(x+1))
then the limit does not exist