Question

How do you find the limit of `(x-5)/(ln(x-4))` as x approaches 5?

Answer 1

If you try to substitute directly `x=5` you get the indeterminate form `0/0`.
Here you can use de L'Hospital Rule. that tells us that:

if `lim_(x->a)(f(x))/(g(x))=0/0` then `lim_(x->a)(f(x))/(g(x))=lim_(x->a)(f'(x))/(g'(x))`

So you can derive both functions and then try for `x=5`. So you get:
`lim_(x->5)(x-5)/(ln(x-4))=` derive both and set `x=5`:
`lim_(x->5)(1)/(1/(x-4))=1`