How do you find the limit of (cos x)^(1/x^2) as x approaches 0?

1 Answer
Jun 3, 2016

1/sqrt(e)

Explanation:

[1]" "lim_(x->0)(cosx)^(1/x^2)

This is an indeterminate form of the type 1^oo. You need to first convert it to the form 0/0 or oo/oo so you can use L'Hopital's Rule. We can do this by using e and ln.

[2]" "=lim_(x->0)e^ln[(cosx)^(1/x^2)]

[3]" "=lim_(x->0)e^[(1/x^2)ln(cosx)]=lim_(x->0)e^[ln(cosx)/x^2]

We can take out e.

[4]" "=e^(lim_(x->0)ln(cosx)/x^2)

This is now an indeterminate form of the type 0/0. We can use L'Hopital's Rule now. Get the derivatives of both the numerator and denominator.

[5]" "=e^(lim_(x->0)(-sinx/cosx)/(2x))=e^(lim_(x->0)(-sinx/(2xcosx))

This is still indeterminate so you must apply L'Hopital's Rule again.

[6]" "=e^(lim_(x->0)(-cosx/(2(-xsinx+cosx)))

You can now get the limit by substitution.

[6]" "=e^(-cos0/(2(-0sin0+cos0)))

[7]" "=e^(-1/2)

[8]" "=color(blue)(1/sqrt(e))