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Proving $\lim_{x\to 10} \ln(x - 3) \neq \infty$ by epsilon-delta

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I am not good at those proofs, and I cannot find any resource because the part of limits is always reduced to the minimum, with few explanations.

$$\lim_{x\to 10} \ln(x - 3) \;=\; +\infty$$

This is clearly false.

$$\forall M > 0, \exists \delta_M > 0 : 0 < |x-x_0|< \delta \implies f(x) > M$$

I am not able to follow this definition to arrive a contradiction, which is what I want.

$f(x) > M$ implies $x > 1 + e^M$.Now I cannot understand how to choose some $\delta$ to show that it's not true that for every $ M > 0$ I have $\ln(x-3) > M$.

How to reason? I don't want to use the negation of the definition.


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