I am asked to evaluate this limit(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\lim_{x\rightarrow \infty}\frac{\sqrt{3x^2+11x+7}+\sqrt{3x^2-11x+7}}{x+\sqrt{x^2+7}} [/tex]

but without use of the basic operations on limits (ex: the limit of a sum is the sum of the limits if they exist, etc.) (because they are to be introduced only in the next chapter). I know the limit is very easily computed: just divide the numerator and the denominator by x and the answer is [itex]\sqrt{3}[/itex]:

[tex]= \lim_{x\rightarrow \infty}\frac{\sqrt{3+\frac{11}{x}+\frac{7}{x^2}}+\sqrt{3-\frac{11}{x}+\frac{7}{x^2}}}{1+\sqrt{1+\frac{7}{x^2}}} = \sqrt{3}[/tex]

Butwhat is the "method" to arrive to this conclusion?The facts that

[tex]\lim_{x\rightarrow \infty}\frac{11}{x} = 0[/tex]

and that

[tex]\lim_{x\rightarrow \infty}\frac{7}{x^2} = 0[/tex]

are useless if we cannot use the operations on limits. I hope I made clear where my confusion comes from. Thanks.

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# Confused about finding limit

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