Q: Find the one-sided limit (if it exists):
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)]
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:
limx→-1(x+1)/(x4-1) = -1/4
Since the limit exists as both the right-handed and left-handed limit, it follows that limx→-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.”