Limit problem

[mjfuentes85]

heres a limit problem I tried working through and i am not sure if i am doing this correct. heres the equation :
lim x -> o+ ((cos x)^(1/x^2))

heres what i started doing:
let y = ((cos x)^(1/x^2))
took the natural log of both sides to get ride of the exponent on the right side of the equation:

im x -> o+ ln y = im x -> o+ ((ln cos x) / x^2)
I am not sure if theres a better way to show the equation on this post. Any ideas for my next step ??
-Marcus

 

[lookagain]

heres a limit problem I tried working through and i am not sure if i am doing this correct. heres the equation :
lim x -> o+ ((cos x)^(1/x^2))

heres what i started doing:
let y = ((cos x)^(1/x^2))
took the natural log of both sides to get ride of the exponent on the right side of the equation:

im x -> o+ ln y = im x -> o+ ((ln cos x) / x^2)
I am not sure if theres a better way to show the equation on this post. Any ideas for my next step ??
-Marcus

\(\displaystyle Instead,\)

\(\displaystyle \ let \ y \ = \ \lim_{x \to 0^+}[\cos(x)]^{\frac{1}{x^2}\)

\(\displaystyle ln(y) \ = \ ln\{\lim_{x \to 0^+}[\cos(x)]^{\frac{1}{x^2}}\}\)

\(\displaystyle ln(y) \ =\ \lim_{x \to 0^+}ln[\cos(x)]^{\frac{1}{x^2}\)

\(\displaystyle ln(y) \ = \lim_{x \to 0^+}\frac{1}{x^2}ln[\cos(x)]\)

\(\displaystyle ln(y) \ = \lim_{x \to 0^+}\frac{ln[\cos(x)]}{x^2}\)


Further strategy:

The right-hand side is an indeterminate form.

1) Use L'Hopital's Rule twice on the right-hand side so that the
denominator winds up being a constant, and after that,

2) have the base e raised to each side, so that the left-hand side
becomes y, and the right-hand side will be the base, e, raised to
a constant.