25: After substituting in the limit point you should find
that the fraction in the tangent is of the type "infinity over infinity", but
fortunately we know that when x 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
![](gif7/eeb7ay1.gif)
Note than when substituting
π/2
into the tangent, we had to check from which side this number is
approached. Since
x/(x + 1) < 1 for
positive x, 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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