Question

lim (x^2+4)/(x+1)
x->-1

Answer 1

x^2+4 is ALWAYS positive (since x^2 is either 0 or positive and 4 is positive)


So the sign of (x^2+4)/(x+1) will depend entirely on the denominator x+1


Notice that if x > -1, then x+1 > 0. So if x > -1, then the entire expression is positive.


But if x < -1, then x + 1 < 0, which means that the entire expression will be negative.


So what's going on here is that as x moves towards -1 from the left, f(x) is becoming a smaller negative. When it hits -1, there's a division by zero error which causes a discontinuity. After it moves on and increases, f(x) then becomes positive.


So this sudden jump from negative to positive (as x passes -1 from left to right) means that there is a jump discontinuity. This analysis allows us to conclude this without even graphing the function. Of course, you can graph to confirm this if you want.


All this means is that the limit as x --> -1 does not exist. This is because the left and right hand limits are not equal.