1. Limits and Continuity

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

Determine the following limits at infinity.

lim x→ ∞ x^−6

Determine the following limits at infinity.

lim x→−∞ x^−11

Determine the following limits at infinity.

lim t→∞ (−12t^−5)

Determine the following limits at infinity.

lim x→∞ (3+10/x^2)

Determine the following limits at infinity.

lim x→∞ (5 + 1/x +10/x^2)

Determine the following limits at infinity.

lim t→∞ et,lim t→−∞ e^t,and lim t→∞ e^−t

Sketch a possible graph of a function f that satisfies all of the given conditions. Be sure to identify all vertical and horizontal asymptotes.

$f(-1)=-2$, $f\left(1\right)=2$, $f\left(0\right)=0$, $\underset{x\to \infty }{\lim }f\left(x\right)=1$, $\underset{x\to -\infty }{\lim }f\left(x\right)=-1$

Explain the meaning of lim x→−∞ f(x)=10.

Estimate lim θ→0 sin 2θ / sin θ using the graph in part (a).