Calculation of a limit with the De L'Hopital rule

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Exercise

Calculate the following limits of fig. 1.

Solution
Here we have the indeterminate form "infinite - infinite":

It follows:

From the previous exercise:

That is, the indeterminate form "zero for infinity" (0 oo) which can be traced back to the indeterminate form "zero on zero" (0/0):

By changing the variable t=1/x we obtain

We apply the De L'Hopital rule

To calculate the numerator derivative, we apply the logarithmic derivative method:

But

It follows:

Conclusion: