can someone help me find the limit of
1. lim ((sin x)/ (pi -x)) as x to pi
2. lim ((x+1)ln x/ sin x) as x tend to 0 from above
3. lim xe^(1/x) as x tends to infinity.
I think "challenging" may be pushing it.
1.Notice that $\displaystyle \lim_{x\to\pi}\frac{\sin(x)}{\pi-x}=-\lim_{x\to\pi}\frac{\sin(x)-\sin(\pi)}{x-\pi}=-\bigg[\sin(x)\bigg]'\bigg|_{x=\pi}=1$
2. $\displaystyle \lim_{x\to0}\frac{\ln(x)(x+1)}{\sin(x)}$. Merely note that on a small enough neighborhood of $\displaystyle 0$ this is clearly bigger than $\displaystyle \ln(x)$ (or is it?...what can you say?)
3. $\displaystyle \lim_{x\to\infty} xe^{\frac{1}{x}}=\lim_{x\to0^+}\frac{e^x}{x}\to\in fty$