Asymptotically Equal Real Functions/Examples/x and x+1
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Example of Asymptotically Equal Real Functions
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = x + 1$
Let $g: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map g x = x$
Then:
- $f \sim g$
as $x \to +\infty$.
Proof
Since:
- $\dfrac {\map f x} {\map g x} = 1 + \dfrac 1 x$
we have:
- $\ds \lim _{x \mathop \to +\infty} \dfrac {\map f x} {\map g x} = 1$
$\blacksquare$