Let $y_1(x),y_2(x)$ be the solutions of $y''+p(x)y'+q(x)y=0$ and satisfy that $y_1(x_0)=y_2(x_0)=0$，$y_1(x) \neq 0$ where $p(x),q(x)$ are continuous over $(-\infty,+\infty)$ and $x_0$ be some real number. Prove there exists a constant $C$ such that $y_2(x)=Cy_1(x)$.  Since at $x_0$ we have $y_1(x_0)=y_2(x_0)=0$ Then $W(y_1,y_2) = 0 \Rightarrow y_1 $ and $y_2$ are linearly depedendent $\Rightarrow$ $y_2(x)=Cy_1(x)$. This will work?
Suppose that $y_1(x)$ and $y_2(x)$ are solutions of the differential equation $y''+py'+qy=0$ on $I$. How can I show that if $y_1$ and $y_2$ vanish at the same point then they are not linearly independent?   Here is my attempt to prove the problem. Since $y_1(x)$ and $y_2(x)$ are solutions of the differential equation $y''+py'+qy=0$ on $I$, it is enough for me to show that the Wronskian of $y_1$ and $y_2$ denoted by $W(y_1,y_2)$ is zero. Now since $y_1$ and $y_2$ vanish at the same point say $p$ we have $y_1(p)=0$ and also $y_2(p)=0$.   Solving for $W(y_1,y_2)(p)$ we have: $y_1(p)y_2'(p)-y_2(p)y_1'(p)=0$ since $y_1(p)=0$ and also $y_2(p)=0$.   Am I correct? Thanks
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Thanks $\lim_{n\to\infty}a_nb_n=? $ given that $\lim_{n\to\infty}a_n=0$ and $\lim_{n\to\infty}b_n=\infty$ A. $\lim_{n\to\infty}a_nb_n=?$ given that $\lim_{n\to\infty}a_n=0$ and $ \lim_{n\to\infty}b_n=\infty$  B. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=0$ and $\lim_{n\to\infty}b_n=0$  C. $\lim_{n\to\infty}{a_n \over b_n}=?$ given that $\lim_{n\to\infty}a_n=\infty$ and $\lim_{n\to\infty}b_n= \infty$    These seem to be difficult situations I come across when I have to evaluate limits.   Some examples,    $\lim_{n\to\infty}n^a\sin(1/n^b)$ with $a,b&gt;1$   $\lim_{n\to\infty}{n^a\over \sin(1/n^b)}$ with $a,b&gt;1$   $\lim_{n\to\infty}{1/n^a\over \sin(1/n^b)}$ with $a,b&gt;1$   $\lim_{n\to\infty} nf(x_n)$ [where we only know that $\lim \ f(x_n) = 0]$    My first question is : Are the two situations  A and B  equivalent ? 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Here's the loop I'm using:  &lt;?php $paged = get_query_var('paged') ? get_query_var('paged') : 1; $args  = array(     'posts_per_page' =&gt; 4,     'orderby'        =&gt; 'modified',     'post_type'      =&gt; "projekte",     'paged'          =&gt; $paged ); $custom_query = new WP_Query($args); $c = 0; while($custom_query-&gt;have_posts()) :     $custom_query-&gt;the_post();     ?&gt;     &lt;article&gt;...&lt;/article&gt;     &lt;?php endwhile; wp_reset_postdata(); // reset the query ?&gt;   I've gone through the functions.php a million times looking for a some limiting get_pre_posts filter but there's nothing I can see.  I also got an error log setup and nothing is returned there, I've tested it that it works.  Any suggestions? I noticed the other day when serving the public that when asking for something, people were saying "Can I get an xyz, please". The previous time I had such a job it was "Could I have an xyz", or "May I have..." if they were being very polite.  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