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I know this problem is not really a statistical problem but more a math problem, but I am sure you can help me though:

I have a probability function dependend on time $t$:

$$p(t)=\frac{1}{e^{at+\ln(2)}}+0.5$$

(the factor a is just a scaling factor)

So if I plot this function, I can see, that for $t$ tending to infinity the probability approx is $0.5$. So now I want to show this in a mathematical way, so I thought the limes would be appropriate for this case? Is this true? So how can I apply lim in this case (I am not good in math I know). Thanks.

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2 Answers

As $\lim_{t\to\infty} [at+\ln(2)]=\infty$ and $x\mapsto e^{x}$ is a continuous function, then also $\lim_{t\to\infty} [e^{at+\ln(2)}]=\infty$. But then $$ \lim_{t\to\infty}\frac{1}{e^{at+\ln(2)}}=0 $$ and hence we conclude that $$ \lim_{t\to\infty}p(t)=\lim_{t\to\infty}\frac{1}{e^{at+\ln(2)}}+0.5=0+0.5=0.5. $$

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I think by limes you mean the limits.

And you would simply ask which is the limit of your function i.e.

\begin{equation} \lim_{t \to \infty} p(t)\\ \lim_{t \to \infty} \frac{1}{e^{at+ln(2)}}+0.5 \end{equation}

Since any number elevated to the infinity equals infinity, by definition, the inverse of infinity is zero.

So at the limit, you only will have the 0.5

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