What is the symbolic form of "there does not exist a largest natural number " Other students in office hour said this is the correct form
$(\forall x)(\exists y)(y>x)$
{ for all x natural number, there exists y such that y is greater than x }
But
"there does not exist a largest natural number "
$\neg(\exists x)(x\text{ largest natural number})$
Am I even close ? 

It depends on whether with largest natural number you mean a greatest or a maximal element, though the difference would be important only for a non-total order.
If $(X,<)$ is a partially ordered set, we say that $a\in X$ is


*

*a greatest element  if $\forall x\in X\colon (x< a\lor x=a)$

*a maximal element if $\forall x\in X\neg(a<x)$


(In a totally ordered set, exactly one of $x<a$, $x=a$, $x>a$ must be true, hence therre the notions of greatest and maximal element coincide).
Hence "There is no $a$ such that $a$ is a greatest/maximal element" translates "literally" to either
$$ \neg\exists a\in \Bbb N\colon \forall x\in\Bbb N\colon (x<a\lor x=a)$$
or
$$ \neg\exists a\in \Bbb N\colon \forall x\in\Bbb N\colon \neg(a<x)$$
If you make use of "$\neg\exists=\forall\neg$" and "$\neg\forall=\exists\neg$", you might equivalently write
$$ \forall a\in \Bbb N\colon \exists x\in\Bbb N\colon (x\not <a\land x\ne a)$$
and
$$ \forall a\in \Bbb N\colon \exists x\in\Bbb N\colon (a<x),$$
respectively.
