Limit of x4 + 5x3 + 6x2 over x2 (x+1) - 4 (x+1) as x approaches 2

Question

Thanks in advance. Please help me.

Answer 1

Top and bottom factorise: x^2(x+2)(x+3)/(x^2-4)(x+1)=

x^2(x+2)(x+3)/(x-2)(x+2)(x+1)=

x^2(x+3)/(x-2)(x+1).

However, when x=2 the denominator goes to zero. When x=2 the numerator goes to 20, a finite number. The limit as x approaches 2 is therefore infinity (dividing by zero).

The limit as x approaches -2 is 4/(-4)(-1)=1, and perhaps this is what the question should be. The expression has a removable anomaly which allows the limit to be evaluated when x=-2.