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$x=O(x+5)$ True or False?

I think

$lim_{x \to \infty} \frac{x+5}{x}=1$, both the functions are growing at the same rate, so is $x=O(x+5)$ false.

But the solution is

true; $x < x + 5 \Rightarrow \frac{x}{x+5}< 1 $ if $x > 1$ (or sufficiently large)

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    If two functions are growing at the same rate, they indeed should be O() of each other. – 2017-02-10
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    I would like to know how you came to the conclusion that the statement is false. Actually you are showing a sufficient condition for it to be true and then just say that it is false without any explanation. I would like to understand that. – 2017-02-10
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    @LeBtz, my laziness....,didn't read big oh carefully :(, thank you – 2017-02-11

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The big-O notation used for example to indicate

$$f(z) = \mathcal{O}g(z) ~~~~~~~~~~~~~ z\to z_0$$

means that $f(z)$ is asymptotically bounded by $g(z)$ in magnitude, as $z\to z_0$.

$$|f(z)| \leq C|g(z)|$$

So, is their ratio bounded in magnitude?

$$\left| \frac{x}{x+5} \right|$$

Hint: they are both linear functions.