Limiting magnitudes for different telescopes
Your calculated estimate may be about correct for the limiting magnitude of stars, but lots of what you might want to see through a telescope consists of extended objects-- galaxies, nebulae, and unresolved clusters.  The magnitude estimates of these objects tend to consider them as point sources, but as extended objects they have much less of a contrast difference with the sky.  In my experience in an equally light-polluted location with a similar-size telescope, the sky brightness makes the limiting magnitude of extended objects between 7 and 8, even with a light-pollution reduction filter in use.
That isn't to say a telescope is worthless in the city; you can still observe the Moon and planets, resolved clusters are pretty, and you can observer the brighter extended objects.  But bringing the telescope to a dark-sky location makes an incredible difference with many more objects visible.