Damsel in Distress

[LaraDeSoleil]

This is going to sound stupid.

I am a sixteen year old girl looking at a simple inequality limits problem. THE problem is that I don't agree that the textbook answers are consistent, and wouldn't you know it, such a simple thing, I've not been able to verify my suspicion anywhere, and I don't want to look more stupid than I am in class.

So here goes. Hopefully someone out there will take pity on a young woman pulling the blond hair out of her silly head.

The first inequality, goes like this:

Find the limits in the following case:

Let f(x) = x^2 - 5 < or = 3, x is < or = 3

Second line of inequality: x + 2, x > 3

Find the limit if (a) limit is 3 minus sign (left side approach) (b) limit is 3 plus sign (right side approach) (c) limit = 3 (both sides considered)

The book says that a and b have limits, but the third does not because the limit of 3 from the right side is different from the value of 3 from the left.

The answers given are: (a) 4 (b) 5 (c) no limit exists

So far so good.

However, the very next problem gives the very same sort of situation, and, this time, according to the book, it doesn't matter that the limits from each side of three is different.

This problem states:

Let f(x) = x^2 - 5, x < or = 3

Second line of inequality: x + 1, x > 3

Find the limit if (a) limit is 3 minus sign (from the left) (b) limit is 3 plus sign (from the right) (c) limit = 3 (consider both sides)

This time the book says all is well in all three cases, even (c). The answers given are (a) 4 (b) 4 (c) 4

It seems to me that both situations require the same response. That is (c) either should be no limit because the value of three varies from the right and left side OR they both should have a limit.

What am I missing?

I've tried to see a difference other than the x + 2 in the first, and the x + 1 in the second, but that's all I see, and that shouldn't make a difference in terms of determining the limits, should it?

Damsel in Distress, Lara

 

[daon]

Firstly, something like "I'm 16 year old blonde girl in distress" doesn't need to be said. In fact, that you mention it at all leads me to believe the opposite, but it doesn't matter.

Now, if a and b agree, c will have to be the same as them. That is the definition of a real limit existing; that the left and right limits exist and agree. One thing to note is that this piecewise function is continuous at 3 (and everywhere else), hence its limit will exist everywhere and be equal to its value.