Huge headache on evaluating floor function limits
I have a few questions regarding floor functions and how to evaluate their limits:
So far, when asked to evaluate the limit of an expression's floor function, I would simply get the floor function of x and substitute this value for the expression.
For example given the problem:
lim [_x+1_] as x-> 3 from the left,
I know the function splits, giving me
2 <= x < 3 (approaching 3 from the left)
3 <= x < 4 (approaching 3 from the right)
therefore, I take 2 and plug it in to the function, giving me a result of 3. This is consistent with wolfram alpha's answer
However, I get the wrong answer when I do the same exact thing in:
lim [_2x-1_] as x-> -2 from the left
What I do is like what I have shown above, that is:
-3 <= x < -2
-2 <= x < -1
So I take -3, plug it in to (2x-1) and get (2)(-3) - 1= (-6) - 1 = -7 when the correct answer is given as -6.
The method I'm using works when I have to get the limit as x approaches -2 from the right though. So given these kinds of problems, what's the proper way of evaluating floor functions?
On the other hand, when I'm given a function
lim [_1-x_] as x-> -1 from the left, I get the intervals
-2 <= x < -1
-1 <= x < 0
Plugging in the value of -2 yields 3 when the correct answer is 2 while solving for the same function as x approaches -1 from the right should yield 1, but plugging in -1 will result in 2.
I was able to evaluate the limit as x-> -1 from the left by doing it this way:
-2 <= 1-x < -1
-3 <= -x < -2
3 > x >= 2
and getting the consistent result of 2
But if I do the same procedure to any of the previous examples, I get the wrong result. For example:
lim [_x+1_] as x-> 3 from the left
2 <= x+1 < 3
1 <= x < 2
Why does this happen?
lim [_2x-3_] as x->-2 from the left
According to wolfram alpha the answer is -8
Now I'm really confused as to how this answer came up. I can get -9 via
-3 <= x < -2, getting -3 and substituting 2(-3) - 3 = -9. Where did -8 come from?
Any help regarding any of my questions would be greatly appreciated, especially with regard to the proper steps in evaluating floor functions in general.
On another note, is it possible that wolfram alpha is wrong?