Evaluate the limit.

Question

lim (cos(x))^(7/x^2) as x goes x-->0^+

Answer 1

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