Can someone solve this problem?

Question

Lim ( 7/ln(x-6) - 7/x -7)
x->7

L'H rule is required I guess, I've spentnan hour on it already. Please help the answer is 7/2 but I can't seem to get it.

Answer 1

L'Hopital's rule is the way to go. First, you need to write the limit as a ratio, so get the two terms over a common denominator:

lim 7[(x-7) - ln(x-6)]/[ln(x-6) (x-7)]
x->7

Now we have the indeterminate form 0/0 when x=7.

One application of L'Hopital's Rule gives

lim 7[(1 - 1/(x-6)]/[ln(x-6) + (x-7)/(x-6)]
x->7

Multiply top and bottom by (x-6) and simplify:

lim 7[(x-7)]/[(x-6) ln(x-6) + (x-7)]
x->7

Still of the form 0/0, so use the Rule again:

lim 7(1)/[1 + ln(x-6) + 1]
x->7

Now you can evaluate.