1. Limits and Continuity

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

e. $\underset{x\to \frac{\pi }{2}}{\lim }\cot x=0$ . (Hint: Graph y=cot x)

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 $p\left(t\right)=\frac{3500t}{t+1}$.

Given the graph of $f$, find a number $\delta$ such that if $0<|x-3|<\delta$, then $\left|f\right(x)-2|<0.5$.

Use series to evaluate the limit: $\underset{x\to 0}{lim}\frac{1-\cos \left(2x\right)}{1+2x-e^{2x}}$.

Approximate the sum of the series $\sum _{n=1}\infty \frac{{-}^{n}6n}{n!}$ correct to four decimal places.

Which of the following functions is continuous on the interval $(0,1)$?

Use the graph of f in the figure to evaluate the function or analyze the limit. <IMAGE>

lim x→3^+ f(x)

For which positive integers $k$ is the following series convergent? $\sum n_1\infty \frac{{(n!)}^2}{(kn)!}$

Given that $f^{\left(n\right)}\left(0\right)$ = $(n+1)!$ for $n=0,1,2,\dots$, which of the following is the Maclaurin series for $f\left(x\right)$?

A tangent line approximation of a function value is an overestimate when the function is:

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limh→0− h / sin 3h

Which of the following statements about the function y=f(x) graphed here are true, and which are false?

i. f(0)=1

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)

lim x → ±∞ f(x) = 0, lim x → 2⁻ f(x) = ∞, and lim x → 2⁺ f(x) = ∞

Complete the following steps for the given functions. 

a. Find the slant asymptote of $f$.

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

Find the radius of convergence, $r$, of the series $\sum \underset{}{n=1}\stackrel{}{\infty }\frac{{(-1)}^n{x}^n}{5^n}$.