answer:Given the limit: [ lim_{x to 0} x cot(x) ] First, recall that (cot(x)) is the reciprocal of (tan(x)), so we can rewrite the expression as: [ lim_{x to 0} frac{x}{tan(x)} ] To evaluate this limit, we apply L'Hopital's Rule, which states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives: [ lim_{x to 0} frac{1}{sec^2(x)} ] Now, we substitute (x = 0) into the simplified limit: [ frac{1}{sec^2(0)} ] Since (sec(0) = 1) (because (cos(0) = 1)), the limit becomes: [ frac{1}{1^2} ] Thus, the limit is: [ 1 ] Therefore, the solution is: [ boxed{lim_{x to 0} x cot(x) = 1} ]

answer:To prove (A cup B)' = A' cup B', we need to show that (A cup B)' subseteq A' cup B' and A' cup B' subseteq (A cup B)'. (A cup B)' subseteq A' cup B': Let x in (A cup B)'. Then, for every neighborhood U_x of x, U_x backslash {x} cap (A cup B) neq emptyset. This means that either U_x backslash {x} cap A neq emptyset or U_x backslash {x} cap B neq emptyset. Therefore, either x in A' or x in B'. Hence, x in A' cup B'. A' cup B' subseteq (A cup B)': Let x in A' cup B'. Then, either x in A' or x in B'. In either case, there exists a neighborhood U_x of x such that U_x backslash {x} cap A neq emptyset or U_x backslash {x} cap B neq emptyset. Therefore, U_x backslash {x} cap (A cup B) neq emptyset. Hence, x in (A cup B)'. Therefore, (A cup B)' = A' cup B'.

answer:Liquidity refers to the ease with which assets can be converted into cash. Stocks and bonds are highly liquid as they can be converted into cash within days. However, large assets like real estate, plants, and equipment are not as easily converted. Out of the options provided, a corporation (referring to corporate investments, not corporate finance) generally has the least liquidity. While mutual funds and checking accounts can be easily redeemed or accessed for cash, corporate investments such as stocks or bonds in private companies may have limited trading and could take longer to sell, thus making them less liquid. In corporate finance, the focus is on managing funding sources, a company's capital structure, decisions to enhance shareholder value, and the financial tools and analysis used to allocate resources.

answer:To find the derivative of the given equation, we can use the product rule, which states that if we have two functions f(x) and g(x), then the derivative of their product f(x)g(x) is given by: frac{d}{dx}(f(x)g(x))=f'(x)g(x)+f(x)g'(x) In our case, we have f(x)=x^2 and g(x)=tan 2x. So, we can find the derivative of each of these functions: f'(x)=frac{d}{dx}(x^2)=2x g'(x)=frac{d}{dx}(tan 2x)=2sec^2 2x Now, we can apply the product rule to find the derivative of y=x^2tan 2x: frac{dy}{dx}=frac{d}{dx}(x^2tan 2x) =2xtan 2x+x^2(2sec^2 2x) =2xtan 2x+2x^2sec^2 2x Therefore, the derivative of y=x^2tan 2x is 2x^2sec^2 2x+2xtan 2x. The answer is frac{dy}{dx}=2x^2sec^2 2x+2xtan 2x