Question

What is the limit of lnx-ln(pi)/x-(pi) when x approaches to (pi) from left?

Answer 1

`lim_(xrarrpi^-)(ln(x)-ln(pi))/(x-pi)=1/pi`

Explanation:

`lim_(xrarrpi^-)(ln(x)-ln(pi))/(x-pi)`

Note that this fits the form for the limit definition of the derivative at a point:

`f'(a)=lim_(xrarra)(f(x)-f(a))/(x-a)`

So, for `lim_(xrarrpi^-)(ln(x)-ln(pi))/(x-pi)`, we see that `f(x)=ln(x)` and `a=pi`, so this limit is equivalent to `f'(pi)` when `f(x)=ln(x)`.

If `f(x)=ln(x)`, then `f'(x)=1/x` and `f'(pi)=1/pi`. So

`f'(pi)=lim_(xrarrpi)(f(x)-f(pi))/(x-pi)=lim_(xrarrpi^-)(ln(x)-ln(pi))/(x-pi)=1/pi`

The sidedness of this limit has no effect on the answer.