I'm looking to evaluate
limx→+∞√x2+4x+1−x
The answer in the book is 2. How do I simply evaluate this problem?
I usually solve limits such as this with the short cut method, i.e (Numerator degree < Denominator degree) = 0 ; (Numerator degree = Denominator degree )= take ratio of leading coefficients; (Degree numerator > degree denominator )= take leading terms and use algebra to simplify and then plug in -\infty or +\infty
Please keep in mind that I do not know L'Hopital's rule.
Answer
\begin{align}\lim_{x\to \infty}\sqrt{x^2+4x+1}-x&=\lim_{x\to\infty}(\sqrt{x^2+4x+1}-x)\cdot\frac{\sqrt{x^2+4x+1}+x}{\sqrt{x^2+4x+1}+x}\\&=\lim_{x\to\infty}\frac{(x^2+4x+1)-x^2}{\sqrt{x^2+4x+1}+x}\\&=\lim_{x\to\infty}\frac{4+\frac 1x}{\sqrt{1+\frac 4x+\frac{1}{x^2}}+1}\\&=\frac{4}{1+1}\end{align}
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