Question

How do you determine the limit of `(x-1)/(x^3 +2x^2)` as x approaches -2+?

Answer 1

This is not a detailed, rigorous, formal approach. But if you generalize the reasoning, you can get the correct answer for this kind of limit.

Explanation:

The numerator approaches a non-zero number while the denominator approaches `0`.

That tells us that we have some kind of infinite limit.

So we will factor and analyze the sign of the factor to determine whether the expression is increasing or decreasing without bound (going to `oo` or `-oo`) as `xrarr-2^+`

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

As `xrarr-2^+`,

`x-1 rarr -3`
`x^2 rarr9`

and

`x+2 rarr0^+`

That is, for `x` is approaching `-2` from the right (through values like `-1.8` and `-1.9` and `-1.993`), the factor `x+2` is a positive number that is close to `0`.

So we have the signs: `((-))/((+)(+)) = -`. The result is a 'big negative' number.

`lim_(xrarr-2^+) (x-1)/(x^3+2x^2) = -oo`