One-sided Limit Example

Q:  Find the one-sided limit (if it exists):

lim_x→-1^–     (x+1)/(x⁴-1)

A:  So we need to find the limit of this function (x+1)/(x⁴-1) as x approaches -1 from the left.  Remember, from the left means as x gets closer and closer to -1, but is still smaller.

The concept: What is happening to this function as x = -2, x = -1.5, x = -1.1, x = -1.0001, etc…

We test first and plug -1 into the function: (-1+1)/((-1)⁴-1) = 0/0

Whenever you get 0/0, that is your clue that maybe you need to do “more work” before just plugging in or jumping to conclusions.

So, let’s try “more work” — usually that means simplifying.  I see that the denominator can factor.  We have:

(x+1) / (x⁴-1) = (x+1) / [(x²-1)(x²+1)]

Let’s keep factoring the denominator:

(x+1) / [(x-1)(x+1)(x²+1)]

Now, it appears there is a “removable hole” in the function.  This means, we can remove this hole by reducing the matching term in the numerator with the matching term in the denominator:

(x+1) / [(x-1)(x+1)(x²+1)]

= 1 / [(x-1)(x²+1)]

Notice that hole exists when x = -1 (and it was removable! This is good news for us since we are concerned with the nature of the function as x approaches -1)

Now that we have removed that hole, let’s once again try to plug in -1 to see what we get.

1 / [(-1-1)((-1)²+1)]

1 / [(-2)(2)]

-1/4

So, after removing the hole at (x+1), we found the function value when x=-1 is -1/4.

Due to the nature of this function, this means:

lim_x→-1(x+1)/(x⁴-1) = -1/4

Since the limit exists as both the right-handed and left-handed limit, it follows that lim_x→-1^–     (x+1)/(x⁴-1) must also be -1/4.  It ended up not being necessary that we only do a “one-handed limit analysis.”