Posted by mike on Monday, December 6, 2010 at 7:59pm.
L'Hôpital's rule applies when there is a fraction whose numerator and denominator are both undefined or zero.
Since the given expression is not a fraction, we need to transform it to a form where L'Hôpital's rule applies.
Taking log is a good way when powers are involved:
ln((cos(x))^(7/x^2))
=(7/x²)*ln(cos(x))
=7ln(cos(x))/x²
Now both numerator and denominator become zero as x->x+.
differentiate with respect to x:
7(-sin(x)/cos(x)) / 2x
=-7tan(x)/2x
Since the expression is still undefined when x->0+, we can differentiate again:
-7sec²(x)/2
=-7/2 as x->0+
So the original limit can be found by raising -7/2 to the power of e, or
Lim x->0+ cos(x)^(7/x²)
= e-7/2
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