L'hospital rule
posted by mike on .
Evaluate the limit.
lim (cos(x))^(7/x^2) as x goes x>0^+

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}