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1.$\ $Evaluate the limit $$\lim_{n\to\infty}(\sqrt{n^2+3n}-n)$$ 2. $\ $Evaluate the limit $$\lim_{x\to-\infty}(\sqrt{4x^2-2x}+2x)$$

These two questions I really don't have any idea how to even start off. Taking the the variable out of the root doesn't get me any further as well. Especially the second one confuses me as it tends to negative infinty. Help me please.

Andy
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  • See https://math.stackexchange.com/questions/2214526/find-lim-x-to-infty-sqrtx29x3 OR https://math.stackexchange.com/questions/2079417/why-is-lim-limits-x-to-infty-frac1-sqrtx22x-x-equal-to-0/2079432#2079432 – lab bhattacharjee Jul 07 '17 at 12:28
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    Multiply the numerator and denominator by conjugate. Take out common factors and see what happens. – bikalpa Jul 07 '17 at 12:35
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    Suggestion for "... I really don't have any idea how to even start off.": Usually one could look for similar problems first in one's calculus textbooks, unless one is in a terrible school that examples are never taught (well then it's time to read the book). –  Jul 07 '17 at 13:09

3 Answers3

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Hint:

$$\sqrt{a} - b = (\sqrt{a} - b)\cdot\frac{\sqrt{a} + b}{\sqrt{a} + b}$$

Hint 2:

$$\lim_{x\to-\infty} f(x) = \lim_{y\to\infty} f(-y)$$ if you introduce a new variable $y=-x$.

5xum
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for 1) write $$\frac{\left(\sqrt{n^2+3n}-n\right)\left(\sqrt{n^2+3n}+n\right)}{\sqrt{n^2+3n}+n}$$ for 2) $$\frac{\left(\sqrt{4x^2-2x}+2x\right)\left(\sqrt{4x^2-2x}-2x\right)}{\sqrt{4x^2-2x}-2x}$$

amWhy
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Dr. Sonnhard Graubner
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The simplest solution, we have seen in the other answers, is the trick "rationalize the numerator".

Here is another method: asymptotic expansion. As $n \to +\infty$,

$$ \sqrt{n^2+3n} = n\;\left(1+\frac{3}{n}\right)^{1/2} =n\left(1+\frac{3}{2n}+O(n^{-2})\right) = n + \frac{3}{2} + O(n^{-1}) \\ \sqrt{n^2+3n} - n = \frac{3}{2} + O(n^{-1}) $$ So we get not only the limit $3/2$, but also an estimated rate of convergence.

GEdgar
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