The discussion revolves around calculating the limit of the expression \(\lim_{x\rightarrow \infty} \frac{\sqrt{x^{2}+5} - x}{\sqrt{x^{2}+2} - x}\). The subject area pertains to limits in calculus, specifically focusing on indeterminate forms as \(x\) approaches infinity.
The discussion is ongoing, with participants exploring various algebraic manipulations to address the limit. There is no explicit consensus on a final approach, but some guidance has been offered regarding the use of conjugates and the transformation of terms as \(x\) approaches infinity.
Participants are constrained by the course material, which does not include L'Hôpital's rule, and there are indications of previous errors in the reasoning that have been highlighted for correction.
walker242 said:Homework Statement
Calculate the limit of
<br /> \lim_{x\rightarrow \infty} \frac{\sqrt{x^{2}+5} - x}{\sqrt{x^{2}+2} - x}<br />
lanedance said:how about this, multiply by both conjugates of the numerator & denominator to get:
<br /> \lim_{x\rightarrow \infty} \frac{5 \sqrt{x^{2}+2} + x}{2 \sqrt{x^{2}+5} +x}<br />
already looking in better shape, as its not a difference term that is leading to the zero, which ws the tricky bit, so from here I'd try multiplying through by:
<br /> \lim_{x\rightarrow \infty} \frac{\frac{1}{x}}{\frac{1}{x}}<br />
this should change the terms containing x in the numerator & denominator from tending to infinity, to ones tending to zero...
walker242 said:\lim_{x\rightarrow \infty} \frac{5\sqrt{x^{2}+2} + x}{2\sqrt{x^{2}+5} +x} =
\lim_{x\rightarrow\infty}\frac{5}{2}\cdot \lim_{x\rightarrow\infty}\frac{x\bigg( \sqrt{1+ \frac{2}{x^2}} + 1\bigg)}{x\bigg(\sqrt{1 + \frac{5}{x^2}}+1\bigg)} =
\frac{5}{2}\cdot\frac{2}{2} = \frac{5}{2}