1. Limits and Continuity

Complete the following steps for the given functions. 

b. Find the vertical asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

$f\left(x\right)=\frac{3x^2-2x+5}{3x+4}$

Complete the following steps for the given functions. 

b. Find the vertical asymptotes of f (if any).

$f\left(x\right)=\frac{3x^2-2x+5}{3x+4}$

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

$f\left(x\right)=-3e^{-x}$

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

$f\left(x\right)=1-\ln x$

Use an appropriate limit definition to prove the following limits.

lim x→1 (5x−2) =3;

Use an appropriate limit definition to prove the following limits.

lim x→ 5x^2 − 25 / x − 5=10

Determine the following limits at infinity.

lim x→−∞ 3x^11

Determine whether the following statements are true and give an explanation or counterexample.

a. The graph of a function can never cross one of its horizontal asymptotes.

Determine whether the following statements are true and give an explanation or counterexample.

c. The graph of a function can have any number of vertical asymptotes but at most two horizontal asymptotes.

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 $p\left(t\right)=\frac{2500}{t+1}$.

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 culture of tumor cells is given by $p\left(t\right)=\frac{3500t}{t+1}$.

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 $p\left(t\right)=\frac{1500}{3+2e^{-0.1t}}$.

The hyperbolic cosine function, denoted $\cosh\left(x\right)$, is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\cosh\left(x\right)=\frac{e^{x}+e^{-x}}{2}$.

b. Evaluate $\cosh \left(0\right)$. Use symmetry and part (a) to sketch a plausible graph for $y=\cosh \left(x\right)$.

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→−∞ cot^−1x