Question

помогите решить лимит

Answer 1

$y=(ctgx)^{frac{1}{lnx}}; ; ,; ; lny=ln(ctgx)^{frac{1}{lnx}}=frac{1}{lnx}cdot ln(ctgx)\\limlimits _{xto 0+0}, ln, y(x)=limlimits _{x to 0+0}, frac{ln(ctgx)}{lnx}=Big [; Lopital; ]=\\=limlimits _{x to 0+0}frac{(1/ctgx)cdot (-1/sin^2x)}{1/x}=limlimits _{x to 0+0}frac{-xcdot tgx}{sin^2x}=Big [; tgxsim x; ,; sinxsim x; Big ]=\\=limlimits _{x to 0}frac{-xcdot x}{x^2}=-1; ; Rightarrow \\ limlimits _{xto 0+0}, y(x)=limlimits _{x to 0+0} (ctgx)^{frac{1}{lnx}}=e^{-1}$