In the wake of bruising fights in their own ranks over the "fiscal cliff" and aid for victims of superstorm Sandy - Republicans in the U.S. House of Representatives open a new Congress on Thursday more divided than ever.While their leader, Speaker John Boehner, seems in no danger of losing his position because of the divisions, his ability to speak for his membership in the House appears greatly diminished.That could not come at a worse time for Republicans as they prepare for their next attempt to get more spending cuts out of President Barack Obama. They will try to use the debt ceiling - and Obama's request to raise it - as leverage, as they did in 2011.But if the final days of this Congress were indicative of things to come, Republicans will have a rough time effectively using their majority in the House against Obama, who even Republicans acknowledge is at the top of his game following the Democrat's re-election in November.The fiscal cliff battle to avert steep tax hikes and spending cuts that were due to kick in at the start of this year proved gut-wrenching for Republicans.Obama's demand for a tax hike on the rich challenged a core principle that has guided Republicans for decades: No new taxes. Ever.Yet, late on New Year's night, 85 Republicans in the House did just that, voting to raise income taxes on household income of more than $450,000 a year.