Calculus Calculus Math Forum

 May 29th, 2018, 07:17 AM #1 Member   Joined: Oct 2017 From: Japan Posts: 62 Thanks: 3 How to solve limits (factoring) Hi everyone, if anyone is interested, here is a lesson on how to solve limits by factoring. Please let me know if you have any comment. May 29th, 2018, 08:08 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 Very good! I have just one comment. You refer to the quadratic $\displaystyle x^2+ 3x- 10$ as having roots -5 and 2 and so can be factored as $\displaystyle (x+ 5)(x- 2)$. I think it would be good to state that we know -5 is a root, and that x+ 5 is a factor, of $\displaystyle x^2+ 3x- 10$, because setting x= -5 made the numerator 0. May 30th, 2018, 07:24 AM #3 Member   Joined: Oct 2017 From: Japan Posts: 62 Thanks: 3 Thank you! Even though how to solve quadratics is not the main focus I may indeed have mentioned it. May 30th, 2018, 11:15 AM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 My point is a bit more than that. If the numerator had been a cubic and we are looking at $\displaystyle \lim_{x\to 4}\frac{x^3- 4x^2+ 3x- 12}{x- 4}$ then setting x= 4, $\displaystyle \frac{4^3- 4(4^2)+ 3(4)- 12}{4- 4}= \frac{64- 65+ 12- 12}{4- 4}= \frac{0}{0}$. My point is that the fact that the numerator is 0 when x= 4 tells us that x- 4 is a factor of the numerator. Since 12/4= 3, the factors must be $\displaystyle (x- 4)(x^2+ ax+ 3)= x^3+ ax^2+ 3x- 4x^2- 4ax- 12= x^3+ (a- 4)x^2+ (3- 4a)x- 12$. We must have a- 4= -4 and 3- 4a= 3. a= 0 satisfies both of those: $\displaystyle \frac{x^3- 4x^2+ 3x- 12}{x- 4}= \frac{(x- 4)(x^2- 3)}{x- 4}a$ which, for all x not equal to 4, is equal to $\displaystyle x^2- 3$. The limit, as x goes to 4, is 16- 3= 13. Tags factoring, limits, limitsfactoring, solve Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ABHISHEK MEENA Calculus 5 December 28th, 2012 06:10 AM daigo Calculus 2 June 29th, 2012 04:53 AM shalikadm Calculus 4 May 31st, 2012 09:21 PM oti5 Algebra 5 March 17th, 2012 12:59 PM watkd Algebra 18 September 6th, 2010 02:34 PM

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