Question

Does the x stay or do i change the x into a y?

Answer 1

Given:

`\lim_(x\to1)(sin(\pi x))/(x-1)`

1. You need to evaluate it by substituting this value:

`x=1`

Then:

`=(sin(\pi(1)))/(1-1)=(0)/(0)`

2. Since it has an indeterminate form of zero over zero, you can use L'Hopital Rule and derivate the numerator and the denominator:

- Set up:

`\lim_(x\to1)((d)/(dx)(sin(\pi x)))/((d)/(dx)(x-1))`

- Apply these Derivative Rules:

`\begin{gathered} (d)/(dx)(sin(u(x))=cos(u(x))\cdot u^(\prime)(x) \\ \\ (d)/(dx)(x^n)=nx^(n-1) \\ \\ (d)/(dx)(k)=0 \end{gathered}`

Where "k" is a constant.

Then:

`\lim_(x\to1)(cos(\pi x)\cdot(\pi x)^(\prime))/(1)`
`\lim_(x\to1)(\pi cos(\pi x))/(1)`
`\lim_(x\to1)\pi cos(\pi x)`

3. Evaluate:

`=\pi cos(\pi(1))=\pi(-1)=-\pi`

Hence, the answer is:

`=-\pi`