Mathematical Grammar, Converges to or Converges at, or both? I lost a point on my test because I put after all my math work, "converges at $0$" not "converges to $0$". Should I argue with my teacher about this or is there a reason why I am wrong?
 A: "at" and "to" mean importantly different things here. When you have a sequence, "at" talks about the domain, while "to" refers to the range. If you tell me a sequence coverages at $0$, my immediate thought is that the series evaluated at $0$ converges. However, if you tell me it converges to zero, my immediate thought is that its limiting value is $0$.
Here are some sentences that highlight the distinction. See which of the two words makes more sense in the sentence:
"The value of the function [at/to] $x=0$ is..."
"The function $f:x\to x^2$ maps [at/to] $[0,\infty)$"
A: If $\lim_{x \to a} f(x) = L$ then $f$ converges to $L$ at $a$.
addendum
The following is an exerpt from talkenglish.com
At:
   Used to indicate a place:
      There is a party at the club house.
      There were hundreds of people at the park.
      We saw a baseball game at the stadium.


To
   Used to indicate the place, person, or thing that someone
   or something moves toward, or the direction of something:
        I am heading to the entrance of the building.

  Used to indicate a limit or an ending point:
       The snow was piled up to the roof.
       The stock prices rose up to 100 dollars.

