求一极限：根号下X+P的和乘以X+Q的和（根号完）减去X,X趋向于正无穷

题目

求一极限：根号下X+P的和乘以X+Q的和（根号完）减去X,X趋向于正无穷

答案

lim{√[(X+P)(X+Q)]-X},X→+∞
=lim{√[(X+P)(X+Q)]-X}*{√[(X+P)(X+Q)]+X}/{√[(X+P)(X+Q)]+X},X→+∞
=lim[(X+P)(X+Q)-X²]/{√[(X+P)(X+Q)]+X},X→+∞
=lim[(P+Q)X+PQ]/{√[(X+P)(X+Q)]+X},X→+∞,分子分母同除以X得
=lim[P+Q+PQ/X]/{√[(1+P/X)(1+Q/X)]+1},X→+∞
=[P+Q+0]/{√[(1+0)(1+0)]+1}
=(P+Q)/2