Question: How Plotting function without singularity?

in some of my function i have a big problem which i can't plot thus function even i know what is the shape of plot, i have two type of ploting directly giving parameter and using explor but for this kind of plot we can't use explor so i have to give the function directly parameter but is wasting my time a lot and i can't get my plot even spending a days by changing parameter one by one, my questions is this how i can plot this kind of function without bieng a singular i need this function to be non singular is not importan about the parameter can be any number 
thanks for any help 

plots-long_term_.mw

the shape of plot must be like this but must have two of them 

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