Finding the limit of this function

turbokaz

Homework Statement

limit as x→3 f(x)= 7/ln(x-2) - 7/(x-3)

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.

Homework Helper

Homework Statement

limit as x→3 f(x)= 7/ln(x-2) - 7/(x-3)

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.

You should have gotten 0 for the derivative of the numerator as well. Indicating you would want to take l'Hopital again. Can you show why you didn't?

turbokaz
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.

Homework Helper
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.

Why do you still have an ln in the denominator after taking the next derivative? You aren't showing your work and you are being sloppy about what you aren't showing.

turbokaz
From the original equation then. After cross multiplying, I plug in and get 0/0. After round one of L'hopitals, 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).