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3 $begingroup$ Compute $lim_{x to -1} f(x)$ for a function $f: mathbb R to mathbb R$ such that $$4 = lim_{x to -1} frac{f(x)+2}{x+1} - frac{x}{x^2-1} tag{1}$$ $$ = lim_{x to -1} frac{f(x)+2}{x+1} - frac{frac{x}{x-1}}{x+1}$$ $$ = lim_{x to -1} frac{f(x)+2 - frac{x}{x-1}}{x+1}$$ Solution 1: My approach is that the numerator $f(x)+2 - frac{x}{x-1}$ must approach zero as $x to -1$ because the denominator approaches zero as $x to -1$ and so $lim_{x to -1} f(x) = -frac{3}{2}$ . Are we allowed to do the following, which seems to be the required solution, instead ? This does not seem very rigorous, and I have a feeling there are obvious counterexamples. Of course, we can use $varepsilon-delta$ to check our answer, but I would like to know if and how this can be generalised for any function $f: mathbb R to ...