Question

Qual o limite de f(x)=(πx+7)/x quando x tende a infinito?

Answer 1

Sem l'Hopital:

$\underset{x\to\infty}{lim}~\frac{(\pi x+7)}{x}~.~\frac{\frac{1}{x}}{\frac{1}{x}}~=~\underset{x\to\infty}{lim}~\frac{(\frac{\pi x+7}{x})}{\frac{x}{x}}~=~\underset{x\to\infty}{lim}~\frac{(\pi+\frac{7}{x})}{1}~=~\underset{x\to\infty}{lim}~(\pi+\frac{7}{x})~=\\\\\\=~\pi+\frac{7}{\infty}~=~\pi+0~=~\boxed{\pi}$

Com l'Hopital:

$\underset{x\to\infty}{lim}~\frac{(\pi x+7)}{x}~=~\underset{x\to\infty}{lim}~\frac{(\pi x+7)'}{x'}~=~\underset{x\to\infty}{lim}~\frac{(\pi)}{1}~=~\boxed{\pi}$