The discussion revolves around finding the limit of the function as x approaches 3, specifically the expression 7/ln(x-2) - 7/(x-3). Participants are exploring the application of L'Hôpital's rule due to encountering an indeterminate form.
The discussion is ongoing, with participants providing insights into their reasoning and corrections to their previous steps. Some participants express confusion about their calculations, while others offer clarifications and guidance on the application of L'Hôpital's rule.
There is a recognition of the indeterminate form 0/0 at multiple stages, and participants are reflecting on their approaches to derivatives and simplifications. The original problem context includes specific limits and functions that are central to the discussion.
turbokaz said:Homework Statement
limit as x→3 f(x)= 7/ln(x-2) - 7/(x-3)
Homework Equations
The Attempt at a Solution
Cross multiplied to get it into one quotient: 7x-21-7ln(x-2)/ln(x-2)(x-3). Plugged 3 into get Indeterminate form 0/0. Took derivative of top and bottom then plugged 3 in again and got -6/0. NOT RIGHT. The correct answer is 7/2. I am stuck as to how they got that.
turbokaz said:okay, I see that I should have gotten 0/0 a second time. But I do L'hopital's again and get (7/(x-2)^2)/ln(x-2)+(x-3/x-2) which after plugging in 3 gives me 7/0.
turbokaz said:From the original equation then. After cross multiplying, I plug in and get 0/0. After round one of l'hospital's, I get (7-(7/(x-2)))/ln(x-2)(x-3). Plug 3 in and get 0/0 again. Derivative of the top gives me (7/(x-2)^2) because 7 goes to 0 and by quotient rule of the second term. I used the product rule on the denominator of ln(x-2)*(x-3) and got ln(x-2)+(1/x-2)*(x-3).
turbokaz said:I don't know why I was being so sloppy. You're right, after round 1 I should have that in the denominator, giving way to 0/0 again. Taking the derivative of top and bottom again leads to (7/(x-2)^2)/ 1/(x-2) + 1/(x-2)^2. Plugged in 3 and got the 7/2. I don't know why my work was so messy but whatever, guess I need to SLOW down. Thanks for your patience.