tag:blogger.com,1999:blog-2433841880619171855.post5951603311810721592..comments2024-06-17T08:13:20.015+01:00Comments on Pat'sBlog: The Sharp Edge Between Infinity and "Finity"Unknownnoreply@blogger.comBlogger1125tag:blogger.com,1999:blog-2433841880619171855.post-27269264881994920572011-03-25T16:41:38.928+00:002011-03-25T16:41:38.928+00:00Another way to see this is to recall the linear ap...Another way to see this is to recall the linear approximation sqrt(1 + u) = 1 + u/2 for |u| near zero (the error is roughly x^2/8).<br /><br />For positive values of a and x we have<br /><br />sqrt(ax^2 + ax)<br /><br />= sqrt[(ax^2)*(1 + 1/x)]<br /><br />= x*sqrt(a)*sqrt(1 + 1/x).<br /><br />Thus, for large values of x this is approximately<br /><br />x*sqrt(a)*[1 + 1/(2x)],<br /><br />with an error of approximately<br /><br />x*sqrt(a)*(1/x)^2 / 8<br /><br />= sqrt(a) / (8x)<br /><br />Thus, for large values of x we have approximately<br /><br />sqrt(ax^2 + ax) = x*sqrt(a) + sqrt(a)<br /><br />and, even with the addition of sqrt(a), the result you got is now transparent.<br /><br />By the way, I made a comment in your March 20 entry that others (who might not have thought to check back there) might be interested in.Dave L. Renfrohttps://www.blogger.com/profile/00863074796446784081noreply@blogger.com