Question

How do you find one sided limits algebraically?

Answer 1

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

`lim_{x to 0^-}1/x=1/{0^-}=-infty`

1 is divided by a number approaching 0, so the magnitude of the quotient gets larger and larger, which can be represented by `infty`. When a positive number is divided by a negative number, the resulting number must be negative. Hence, then limit above is `-infty`.

(Caution: When you have infinite limits, those limts do not exist.)

Here is another similar example.

`lim_{x to -3^+}{2x+1}/{x+3}={2(-3)+1}/{(-3^+)+3}={-5}/{0^+}=-infty`

If no quantity is approaching zero, then you can just evaluate like a two-sided limit.

`lim_{x to 1^-}{1-2x}/{(x+1)^2}={1-2(1)}/{(1+1)^2}=-1/4`

I hope that this was helpful.