<s>[INST] Why $\lim_{x\to -1}\frac{x^3-2x-1}{x^4+2x+1}\neq 0$ According to solution, $\lim_{x\to -1}\frac{x^3-2x-1}{x^4+2x+1}=\frac{1}{2}$. Why it is so, when power of polynomial in denominator is greater than in numerator?
 [/INST] If you change: $x+1=t$, then:
$$\lim_{x\to -1}\frac{x^3-2x-1}{x^4+2x+1}=\lim_{t\to 0}\frac{(t-1)^3-2(t-1)-1}{(t-1)^4+2(t-1)+1}=\lim_{t\to 0}\frac{t^3-3t^2+t}{t^4-4t^3+6t^2-2t}=\\
\lim_{t\to 0}\frac{t^2-3t+1}{t^3-4t^2+6t-2}=-\frac12.$$
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