Question

Limit question: lim x-->pi ((e^sinx)-1)/(x-pi)

Answer 1

`\displaystyle\lim_(x\to\pi)(e^(\sin x)-1)/(x-\pi)`

Notice that if
`f(x)=e^(\sin x)`, then
`f(\pi)=e^(\sin\pi)=e^0=1`. Recall the definition of the derivative of a function
`f(x)` at a point
`x=c`:


`f'(c):=\displaystyle\lim_(x\to c)(f(x)-f(c))/(x-c)`

So the value of this limit is exactly the value of the derivative of
`f(x)=e^(\sin x)` at
`x=\pi`.

You have


`f'(x)=\cos x\,e^(\sin x)\implies f'(\pi)=\cos\pi\,e^(\sin\pi)=-1`