8: The fraction in the tangent is of the type "infinity over infinity", but
fortunately we know that when n grows large, the
"+1" part can be
ignored and the fraction can be worked out. Thus the type of the whole
expression can be estimated, it is
![](gif5/eea5ah1.gif)
Note
than when substituting
π/2
into the tangent, we had to check from which side this number is approached.
Since n/(n + 1) < 1, the argument
inside the tangent is always
less than
π/2, so it
approaches this number from the left and the tangent goes to positive
infinity. If
π/2 was
approached from the right, the tangent would go to negative infinity!
We have an
indeterminate power, so
the standard procedure can be applied: Use the "e to ln" trick and
pass to the limit of the expression inside the exponential.
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Answer