New symbol of infinity similar to a 'w' or omega

Based on the original guidance to trim the upper part, one can adjust the .4\LMex clipping amount to suit the desired look.

\documentclass{article}
\usepackage{trimclip,scalerel}
\newcommand\altinfty{\ThisStyle{\clipbox{0pt 0pt 0pt .4\LMex}{$\SavedStyle\infty$}}}
\begin{document}
$\lim x \rightarrow \altinfty$

$\scriptstyle\lim x \rightarrow \altinfty$

$\scriptscriptstyle\lim x \rightarrow \altinfty$
\end{document} 

ADDENDUM

Based on a comment by the OP, an alternative, rather than clipping, is to overlay a strategically placed white rectangle:

\documentclass{article}
\usepackage{xcolor,stackengine,scalerel}
\newcommand\altinfty{\ThisStyle{\stackinset{c}{-.1pt}{t}{-.08\LMex}
  {\textcolor{white}{\rule{1.5\LMex}{.6\LMex}}}{$\SavedStyle\infty$}}}
\begin{document}
$\lim x \rightarrow \altinfty$

$\scriptstyle\lim x \rightarrow \altinfty$

$\scriptscriptstyle\lim x \rightarrow \altinfty$
\end{document} 

Clip three copies of \infty and superimpose them to each other.

\documentclass{article}
\usepackage{trimclip}

\makeatletter
\DeclareRobustCommand{\badomega}{\mathord{\mathpalette\bad@omega\infty}}
\newcommand{\bad@omega}[2]{%
  \ooalign{%
    \clipbox{0pt 0pt {0.75\width} {0.2\height}}{$\m@th#1#2$}\hidewidth\cr
    \hidewidth\clipbox{{0.75\width} 0pt 0pt {0.2\height}}{$\m@th#1#2$}\cr
    \clipbox{0pt {-0.05\height} 0pt {0.55\height}}{$\m@th#1#2$}\cr
  }%
}

\begin{document}

$\badomega\scriptstyle\badomega\scriptscriptstyle\badomega$

\end{document}

This will work also over a colored background.