You can start considering ## \lim_{x\rightarrow 0^{+}} (\cos{x})^{\frac{1}{x}} ##, you have an indeterminate form of this kind ## (1^{-})^{+\infty}##. In general with ## \lim_{x\rightarrow 0^{+}} f(x)^{g(x)}## you can consider ## \lim_{x\rightarrow 0^{+}} e^{\log{ f(x)^{g(x)}}}## and nothing change. The exponential function is monotone and by the properties of ## \log## you have to solve the internal limit:
@goldfish9776, the image you posted is sideways. Please start a new thread with your work shown in the post, not as an image. When you post an image we can't insert a comment at the location of an error -- we have to describe the location.
Also, when you post a sideways image, many helpers will refuse to make the effort to read what you have written, even moreso when the text on the backside of the paper you wrote on also shows on your image. Everything you wrote in longhand can be done using LaTeX. We have a tutorial here: https://www.physicsforums.com/help/latexhelp/ (under INFO --> Help/How-To.