Finding one-sided limits algebraically, without using a graph
How do you do a one sided limit without using a graph, and just doing it algebraically? Like in the example the limit $$ \lim_{x\to 1^+}\frac{x}{x^2-1} $$ how do you find that algebraically?
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$\begingroup$When the denominator approaches zero and the numerator approaches a constant you have a vertical asymptote. So you have the function f: $$f(x) = \frac{x}{x^2-1}=\frac{x}{(x-1)(x+1)}$$ Where there are vertical asymptotes at $x=1$ and $x = -1$. You just need to analyse whether the function is positive or negative close to these values to see if the function is approaching positive or negative infinity.
For the specific limit as $x$ approaches 1+, the numerator will be positive ($x>0$) and the demoniator will be positive ($x-1)(x+1) > 0$ for the range $x>1$, so this limit is positive infinity.
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