Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.


lim x→3 −5x / √4x − 3

If a function f represents a system that varies in time, the existence of lim ${\(\displaystyle\[\lim\)_{t\(\rightarrow\]\infty\)}{f(t)}}$ 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 colony of squirrels is given by $p\left(t\right)=\frac{1500}{3+2e^{-0.1t}}$.

Determine the following limits.

$\underset{x\to 0^{-}}{\lim }\csc \left(x\right)$

Determine $\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\))$ and $\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\))$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{40x^5+x^2}{16x^4-2x}$

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

h(x)=e^x(x+1)^3

Evaluate lim x→0 x + 1/ 1 −cos x.

Determine the interval(s) on which the following functions are continuous. 

p(x)=3x^2−6x+7 / x^2+x+1