This is more of a question regarding the definition of one sided limits rather than finding one sided limits. Say if I was to find the right hand side limit of (x/x+1)(2x+5/x^2+x) as x goes to -2 (from the right). I would substitute in -2 for the x's and find the limit to be 1. Now here that limit would stay the same regardless of which side you are finding it from.
An example of where the limit is different is finding the limit of 1/x from both the left and right hand side as x goes to 0. The left hand side would yield negative infinity where as the right hand side would give positive infinity. So is this the only case where one sided limits actually produce a different limit? When you have a 0 in the denominator?
there could be other function which will give different value of the positive and negative one sided limits. an example will be the piecewise function. for your information. if the denominator is 0, it does not confirm that this limit does not exist. if both the numerator and denominator are 0, the limit may exist. however if the numerator is non zero, but the denominator is zero. then we can conclude that the limit does not exist.