Compute the limit: $$\lim_{z\rightarrow 4} \frac{\sqrt{z}-2}{z-4}$$

This is an example of a limit problem that might be solved in multiple ways. Obviously, both the numerator and the denominator tend to 0 for z approaching 4, resulting in a "0/0" scenario (aka, indeterminate form) inside the limit. This is a case where the application of the L'Hospital rule offers itself. By that rule, the original limit is the same as the limit of a ratio of the derivatives of the original numerator and denominator: $$\lim_{z\rightarrow 4} \frac{\sqrt{z}-2}{z-4}=\lim_{z\rightarrow 4} \frac{\frac{d}{dz}(\sqrt{z}-2)}{\frac{d}{dz}(z-4)}=\lim_{z\rightarrow 4}\frac{\frac{1}{2\sqrt{z}}}{1}=\frac{\frac{1}{2\sqrt{4}}}{1} =\frac{1}{4}$$ The limit is 1/4. If the use of the L'Hospital rule is not desirable (as some teachers require alternative methods in some instances), the alternative method would involve factorization of the denominator and reducing the fraction to a case that does not result in the indeterminate form 0/0.

Identify critical points of the function: $$f(x) = \frac{10}{3}x^3-10x^2-150x+205$$

This function is a cubic polynomial. Critical points of a function are those values of x where its derivative is 0, or does not exist. (Please note that this is one possible definition of a critical point. Some schools include points of inflection, i.e., the second derivative, in this definition as well.) The domain of the polynomial are all Reals and the function exists everywhere, therefore, as the first step, we calculate the derivative (at this point I would check with the student if they know the general rule for a derivative of a monomial, and explain more detail, if necessary): $$\frac{df}{dx}=10x^2-20x-150$$ At the critical points, the derivative is zero: $$10x^2-20x-150=0$$ and that amounts to determining the roots of a quadratic equation. We can use the quadratic root formula, or, as in this case, a factorization scheme as follows: $$10x^2-20x-150=10(x^2-2x-15)= 10(x+3)(x-5)=0$$ (at this point I would check with the student if they know how this factorization was done, and would explain more detail, if necessary) From the above, we can easily identify the roots, namely $$x_1=-3, x_2=5$$ and these correspond to the critical points of the function f(x).

A zoo entrance costs $19 for general public, and $14 for membership subscribers. The 486 tickets sold in one week took in $8629. How many general admission and how many member tickets were sold?

As the first step with this type of problems, we parse the text to identify which variables are going to be the "unknowns" and we assign them a symbol (name). In this case, we are to determine two quantities: the number of general admission tickets sold (let's name that "x"), and the number of membership tickets sold (let that be referred to as "y"). Next step involves setting up the corresponding equation(s). In our case, since there are two unknowns, we are looking to set up two equations. The first equation should reflect the fact that the total weekly revenue of $8629 comes from selling "x" general and "y" membership tickets each at the corresponding price: $$19x + 14y = 8629$$ This by itself would not be sufficient to determine both x and y. But we can add a second equation to capture the remaining bit of information, namely that a total of 486 tickets was sold: $$x + y = 486$$ thus giving us a system of two equations with two unknowns. We are now ready to solve for x, and y. There are multiple methods to solve this, I will be using the substitution method: $$19x + 14y = 8629\\ x+y=486\implies x = 486-y\\ 19(486-y)+14y = 8629\\ 9234-5y=8629\\ y = 121\implies x=486-121=365 $$ We arrived at the solution: There were 365 general (x) and 121 membership (y) tickets sold.