Using the definition of limit, prove $\frac{x+1}{x+2}\to 1$ as $x\to\infty$

Question

How can I show a limit using the mathematical definition? I always computed limits but never learned how to prove it.

$$\lim_{x\rightarrow+\infty}\frac{x+1}{x+2}=1$$

Answer 1

You can divide the top and bottom of the fraction by $x$ and get $\frac{1+\frac{1}{x}}{1+\frac{2}{x}}$.

Answer 2

$$\text{Hint: } \lim_{x \to \infty} \frac{x+1}{x+2} = \lim_{x \to \infty} \frac{x+2 - 1}{x+2} = 1 - \lim_{x \to \infty} \frac{1}{x+2} = \space ...$$

Answer 3

When you say "mathematical definition", I am not sure if you wanted to prove the limit rigorously with an $\epsilon$ argument, but I'll just write it anyway:

Let $\epsilon > 0$ be given. Let $M=\frac 1{\epsilon}$. If $x > M$, then $$\left| \frac{x+1}{x+2}-1\right|=\left|\frac{-1}{x+2}\right|=\frac{1}{x+2}<\frac 1x<\frac 1M=\epsilon.$$