My Math Forum help with finding limit without using L' Hôpital's Rule

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 June 28th, 2017, 05:47 AM #1 Newbie   Joined: Jun 2017 From: Long Beach Posts: 20 Thanks: 0 Hello everybody I appreciate your help with this problem: limit (((1+tan(x))^0.5-(1+sin(x))^0.5)/x^3,x->0) I can solve this with L' Hôpital's Rule by differentiating the numerator and denominator 3 times and finding the limit of the resultant numerator and denominator when x->0 and dividing them: (3/2)/6=1/4! But I'd like to see whether there is another way of solving this problem without using L' Hôpital's Rule, like by some trick or linearization... it was assigned in part of the textbook that hasn't talked about L' Hôpital's Rule up to that point! Thank you for your help in advance... Last edited by skipjack; June 28th, 2017 at 06:26 AM.
 June 28th, 2017, 06:19 AM #2 Global Moderator   Joined: Dec 2006 Posts: 18,951 Thanks: 1599 Hint: multiply numerator and denominator by cos(x)((1+tan(x))^0.5+(1+sin(x))^0.5). Thanks from mgho
 June 28th, 2017, 06:29 AM #3 Senior Member   Joined: Dec 2015 From: Earth Posts: 224 Thanks: 26 $\displaystyle x\rightarrow 0$ $\displaystyle \lim \frac{\sqrt{1+\tan{x}}-\sqrt{1+\sin x}}{x^3}=\lim \frac{\tan{x} - \sin x}{2x^3}=\frac{1}{2} \lim \frac{\sin x(\frac{1}{\cos x}-1)}{x^3}=$ $\displaystyle =\frac{1}{2} \lim \frac{(\frac{1}{\cos x}-1)}{x^2}\cdot \lim \frac{\sin x}{x}=\frac{1}{2} \lim \frac{1-\cos x}{x^2 \cos x}=\frac{1}{2} \lim \frac{1-\cos x}{x^2}=\frac{1}{4}$ Thanks from mgho Last edited by skipjack; June 28th, 2017 at 08:25 AM.
 June 28th, 2017, 06:34 AM #4 Newbie   Joined: Jun 2017 From: Long Beach Posts: 20 Thanks: 0 thanks, but I still don't know what to do with the x^3 that's left in the denominator..
 June 28th, 2017, 06:37 AM #5 Newbie   Joined: Jun 2017 From: Long Beach Posts: 20 Thanks: 0 I got it! thanks a lot idontknow and skipjack
 June 28th, 2017, 08:31 AM #6 Global Moderator   Joined: Dec 2006 Posts: 18,951 Thanks: 1599 (1 - cos(x))/x² = 2sin²(x/2)/(4(x/2)²) = (1/2)(sin(x/2)/(x/2))² Thanks from mgho

 Tags finding, hôpital, hôpitals, hopitals, limit, rule

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