What is $\lim\limits_{x \to \infty} \dfrac{x}{x-1}$?

Question

I want to calculate the limit $$ \lim\limits_{x \to \infty} \dfrac{x}{x-1}. $$ I know that this can be achieved using l'Hospital but cannot figure out how to do this.

Answer 1

You don't have to use L'Hospital: it's a high school theorem that the limit at $\infty$ of a rational function is the limit of the ratio of the highest degree terms of its numerator and denominator.

Answer 2

Like lab bhattacharjee said:

$$\lim_{x\to\infty}\frac{x}{x-1}=\lim_{x\to\infty}\frac{\frac{x}{x}}{\frac{x}{x}-\frac{1}{x}}=\lim_{x\to\infty}\frac{1}{1-\frac{1}{x}}=$$ $$\frac{1}{1-\lim_{x\to\infty}\frac{1}{x}}=\frac{1}{1-0}=\frac{1}{1}=1$$

Answer 3

$$\lim_{x\to\infty}\frac{x}{x-1}=\lim_{x\to\infty}\frac{x-1+1}{x-1}=\lim_{x\to\infty}1+ \frac{1}{x-1}=?$$