Recent content by Vale132

Homework Statement Let S \subset \mathbb{R} be bounded above. Prove that s \in \mathbb{R} is the supremum of S iff. s is an upper bound of S and for all \epsilon > 0 , there exists x \in S such that |s - x| < \epsilon . Homework Equations **Assume I have only the basic proof...

How does that work when R \rightarrow \infty ?

Homework Statement Please let me know if this kind of posting of exact problems from a textbook isn't allowed; if that's the case I'll delete it immediately. From Boas's Mathematical Methods in the Physical Sciences, Third Edition: The Fresnel integrals, \int_0^u sin (u^2)\,du and...

You may be right; I had gotten in the habit from using quotients in earlier parts of the question. I guess "compute" implies that using the chain rule is okay. Thanks for the suggestion. Out of curiosity, would one of the above methods be useful if I had to do it this way?

Or e^{x} = \lim \ \ \ \ \ \ \ (1 + \frac{x}{n})^n \ \ \ ?\\ \ \ \ \ \ \ \ \ n\rightarrow∞ EDIT: Then inserting -1/(x+h) for x, the exponential would go to 1, as would e^(-1/x)?

Do you mean I could use the Taylor series definition for e^x, one centered around 0 and one around h, to turn the exponentials into polynomials? Then the terms without h in the numerator would cancel, and I could use the remaining ones to cancel h in the denominator?

Homework Statement Let f(x) =\begin{cases} 0 & \text{ if } x\leq 0 \\ e^\left ( -1/x \right ) & \text{ if } x> 0 \end{cases} Compute f'(x) for x < 0 and x > 0. Homework Equations f'(x) = \lim \ \ \ \ \ \ \displaystyle{\frac{e^{(-1/(x+h)} - e^{-1/x}}{h}} \\ \ \ \ \ \ \ \ \ \...