Proving $a^b$ is well defined February 11

How do I prove that $$\lim_{(m,n) \to \infty} a_m^{b_n} = a^b$$where $a,b \in \mathbb R$, $a_i,b_i \in \mathbb Q$, $a_m \to a$, $b_n \to b$ and $a$ and $b$ are not both zero.I can prove it with $n$ or $m$ constant, but not when both are limiting at t

Distance between two symmetric equations February 11

I have been requested to solve this problem:Compute the distance between the lines:$L_{1}:\frac{x-2}{3}=\frac{y-5}{2}=\frac{z-1}{-1}$ and $L_{2}:\frac{x-4}{-4}=\frac{y-5}{4}=\frac{z+2}{1}$This is my solution:I specify one point for each line:$P_{L_{1

Division by $x-a$ in a polynomial ring $Rx$ February 11    1

Let $R$ be a commutative ring with identity. Consider the polynomial ring $R[x]$. Suppose $f \in R[x]$ and $a \in R$ are such that $f(a) = 0$. Is it true that $f(x) = (x - a)g(x)$ for some $g \in R[x]$?If $f(x)=\sum_{k=0}^n a_k x^k$, we can write $$f

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