Here is the 4th one:
There is a really useful formula that can help with this. Let's derive it:
First, there is a crucial result that for all functions f:
So, we can write:
This is just e raised to a constant times x.
Taking the derivative:
But now this can be re-written as:
In general, it's better to just remember the formula:
And notice this still holds if b = e:
I thought that whenever the square root sign is written, as in a function definition for example, it means "extract the positive root"... at least it is like that in all the textbooks i've seen. I have never seen any instance where a function definition involving a square root operator was supposed to mean both possible roots.
re: the difference between "undefined" and "indeterminate" -- i think undefined means strictly "there is no meaningful way to define the symbol or operation", so we leave it undefined, 'illegal'. The term 'Indeterminate' is applied only to limit situations, ie, when we have a function like (x-1)/(x^2 - 1) and x approaches 1, the function's value approaches an indeterminate value -- not undefined, but *we are not able to determine the value being approached*.
Yes you can do your question like that too -- except you can see the slope of your line is 2 already, without re-writing it. The key thing to know is: parallel lines have the exact same slope.
So your parallel line's equation will be y = 2x + b ... now you just need to find b. But you know a point on the line, so just substitute the point in this equation!
So the function actually reduces to:
Well we are trying to find
so let's restrict the values of x we are looking at to:
x is in that interval if it is approaching 2, and f(x) = -1 on that interval; it never reaches 2. So when evaluating the limit, f(x) = -1, and the limit is still -1 ... right?
Ok here is another way, no Fourier:
We need to be aware of this formula:
It is evident that this formula always gives -1.
Eg for 2.1 , the floor is 2, the ceiling is 3, and we are always taking the floor - ceiling, which is always going to be -1.
Thus the function simply reduces to f(x) = -1 for all x.
This makes the proof of the limit trivial.