Prove that a horizontal asymptote can never be crossed
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This is a lot simpler of a problem than others posted here, but I was bored in class and decided to work out why a horizontal asymptote exists. Bear in mind that I am still fairly low on the “math ladder.” So to accomplish this I worked off of one example equation that could be made into a general case later on, with the chosen one being $$y = frac{2x+6}{x+1}$$ Now, proving any vertical asymptote is simple (by definition it creates a “divide by zero” error when plugged into the equation), but a horizontal asymptote “proof” requires some manipulation. So, doing a little shuffling of the equation to isolate the variables slightly... $$y = frac{x(2+frac{6}{x})}{x(1+frac{1}{x})}$$ The $x$ on the top and bottom of the function will cancel out, and we are left with $$y = frac{2+frac{6}{x}}{1+frac{1}{x