Properly Defining Limits and Improper Integrals in Calculus

[aaaa202]

I have the expression:

lim_b->-8[½b^2]-lim_b->8[½b^2]

Is it rigorously defined how to calculate this? The question arose because I want the additivity of improper integrals to work and if you take the integral of x from minus infinity to infinity to work the expression above must be zero.

 

[Mandelbroth]

I have the expression:

lim_b->-8[½b^2]-lim_b->8[½b^2]

Is it rigorously defined how to calculate this?

There are no domain restrictions that we need to be aware of here, so we can just say that ##\displaystyle \lim_{b\rightarrow -8}\frac{b^2}{2} - \lim_{b\rightarrow 8}\frac{b^2}{2} = \frac{(-8)^2}{2}-\frac{8^2}{2} = 0##

If you want rigor, consider the ##\epsilon ,\delta## definition of a limit:

$$\lim_{x\rightarrow \alpha}f(x) = \mathfrak{L} \iff \forall\epsilon>0 \ \exists\delta>0:\forall x (0<|x-\alpha|<\delta \Rightarrow 0<|f(x)-\mathfrak{L}|<\epsilon)$$

 

[Bacle2]

In other words, both limits exist, by continuity of f(x)=x^2 , so you can calculate them individually and subtract. Remember that continuity is equivalent to lim_x-->xo f(x)= f(xo) , so if you accept continuity of
f(x)=x^2 ( or prove it in the way Mandelbroth suggested ), the result follows.

 

[aaaa202]

lmao. I now see that what appeared as the number 8 was supposed to be infinity. I think that will change the answer from you a bit .

 

[Mark44]

You can do this:
$$ \int_{-\infty}^{\infty} x~dx = \lim_{b \to \infty} \int_{-b}^b x~dx$$
$$ = \lim_{b \to \infty} \left.(1/2)x^2 \right|_{-b}^b$$
Can you continue from here?