How do you find one sided limits algebraically?

Question

Evelyn Agard · Accepted Answer

Let's examine some examples to show you why you should exercise caution when evaluating a one-sided limit when a quantity is approaching zero because its sign varies depending on which direction it is approaching zero.

#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#.

(Warning: There are no limits when you have infinite limits.)

Here's another illustration that is comparable.

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

One can evaluate the situation as though it were a two-sided limit if no quantity is getting close to zero.

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

I hope you found this useful.

Ezra Adams · Answer

undefined To find one-sided limits algebraically, evaluate the function as the input approaches the desired value from the left or right side. If the function approaches a specific value from both sides, that value is the one-sided limit. If the function approaches different values from the left and right sides, the one-sided limits do not exist.