Homework Statement
Let B be the family of subsets of \mathbb{R} consisting of \mathbb{R} and the subsets [n,a) := {r \in \mathbb{R} : n \leq r < a} with n \in \mathbb{Z}, a \in \mathbb{R} Show that B is not a topology on \mathbb{R}
Homework Equations
The Attempt at a Solution
If B...
Homework Statement
We are given the following Sturm-Liouville eigenvalueproblem:
(p(x)y')' + r(x)y = \lambday
y(-a) = y(a) = 0
on a symmetrisch interval I = [-a, a]. About p(x) and r(x) we are given that p(-x) = p(x) < 0 and r(-x) = r(x) \forallx \in [-a, a]. Show that every eigenfunction...
Homework Statement
Hi
I've been giving the following problem:
We have a differentiable function f: [a,b] \rightarrow \mathbb{R} with f'(a) < 0 en f'(b) > 0. Let c \in \mathbb{R} such that f'(a) < c. Show that there exists a \delta >0 such that for every x \in ]a, a + \delta[ the following...
That's what I was afraid off :(. How would I prove that lim (x,y) → (a,0) e^(x ln y) = 0 and lim z → -∞ e^z = 0? I am familar with the \delta and \epsilon method but I am not sure how to include ∞ in this method.
Homework Statement
Show that lim (x,y) → (a,0) e^(x ln y) = 0 \foralla > 0
Homework Equations
The Attempt at a Solution
I've tried looking at lim (x,y) → (a,0) x ln y seperately.
lim(x,y) → (a,0) x ln y = lim(x,y) -> (a,0) x * lim(x,y) → (a,0) ln y...
Homework Statement
We are given two functions f : \mathbb{R}^n -> \mathbb{R} and g: \mathbb{R}^n -> \mathbb{R}. For every x \in \mathbb{R}^n we define the following:
k(x) = max{f(x), g(x)}
h(x) = min{f(x), g(x)}
The question is:
if lim x-> a k(x) exists and lim x-> a h(x) exists, and...