用定义证明limit x/(x^2-4)=无穷大 x趋近于2

优质解答

任取M>0,取d=min{1,1/(5M)},此时有d<=1,且d<=1/(5M),
当0<|x-2|5M
同时,|x+2|=|x-2+4|<=|x-2|+4<5,1/|x+2|>1/5
|x|=|x-2+2|>=2-|x-2|>1
下面看|x/(x^2-4)|=1/|x-2|*|x/(x+2)|>1/|x-2|*1/5>5M*1/5=M
因此当x趋于2时,原式极限为无穷大. 任取M>0,取d=min{1,1/(5M)},此时有d<=1,且d<=1/(5M),
当0<|x-2|5M
同时,|x+2|=|x-2+4|<=|x-2|+4<5,1/|x+2|>1/5
|x|=|x-2+2|>=2-|x-2|>1
下面看|x/(x^2-4)|=1/|x-2|*|x/(x+2)|>1/|x-2|*1/5>5M*1/5=M
因此当x趋于2时,原式极限为无穷大.