A tiny Exercise: Limit of a Simple Function

A tiny Exercise: Limit of a Simple Function

Recently, somebody asked me, if it is possible, to compute the limit of the following function, when tends towards infinity:

As it turns out, it is possible and actually not that difficult. You can try it yourself and then check the solution afterwards in this post.

We start by writing the term in a slightly different way:

Let us first find an easy limit:

This leads to the conclusion that the numerator

in \eqref{eq:start} becomes:

Since also the denominator has a limit of

we have a situation, where L’Hosptial’s rule can be applied so that:

The derivative of is given as

and the derivative of as:

This finally leads to the simple expression:

Finally, since we silently got rid of the exponential function, we have to insert the found limit into the exponential function and we are done:

Markus Thill

Markus Thill
I studied computer engineering (B.Sc.) and Automation & IT (M.Eng.). Generally, I am interested in machine learning (ML) approaches (in the broadest sense), but particularly in the fields of time series analysis, anomaly detection, Reinforcement Learning (e.g. for board games), Deep Learning (DL) and incremental (on-line) learning procedures.

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