Determine the interval(s) on which the following functions are continuous. 
f(x)=1 / x^2−4

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 x^4=1

Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 

p(0) = 2,lim x→0 p(x) = 0,lim x→2 p(x) does not exist, p(2)=lim x→2^+ p(x)=1

Determine the following limits.

lim θ→π/2 sin^2 θ − 5 sin θ + 4 / sin^2 θ − 1

If a function f represents a system that varies in time, the existence of lim $\underset{t\to \infty }{\lim }f\left(t\right)$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a bacteria culture is given by $p\left(t\right)=\frac{2500}{t+1}$.

Find an interval containing a solution to the equation $2x=\cos \left(x\right)$. Use a graphing utility to approximate the solution.

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$, determine $\underset{x\to a^{+}}{\lim }f\left(x\right)$, $\underset{x\to a^{-}}{\lim }f\left(x\right)$, and $\underset{x\to a}{\lim }f\left(x\right)$.

$f\left(x\right)=\frac{\cos \left(x\right)}{x^2+2x}$