1. Limits and Continuity

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^+ 1 /1 − x=−∞

Let f(x) =x^2−2x+3.

a. For ε=0.25, find the largest value of δ>0 satisfying the statement

|f(x)−2|<ε whenever 0<|x−1|<δ.

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers and assume lim x→a f(x) =L

d. If |x−a|<δ, then a−δ<x<a+δ.

Given the graph of f in the following figures, find the slope of the secant line that passes through (0,0) and (h,f(h))in terms of h, for h>0 and h<0.

f(x)=x^1/3 <IMAGE>

Suppose $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$. Prove that $\underset{x\to a}{\lim }(f\left(x\right)-g\left(x\right))=L-M$.

Suppose ${\displaystyle\lim_{x\to a}}\text{ }f\left(x\right)=L$. Prove that $\underset{x\to a}{\lim }\text{ (}c\left(f\left(x\right)\right)=cL$, where $c$ is a constant.

Use the precise definition of infinite limits to prove the following limits.

$\underset{x\to 4}{\lim }\frac{1}{{(x-4)}^2}=\infty$

Use the precise definition of infinite limits to prove the following limits.

$\underset{x\to -1}{\lim }\frac{1}{{(x+1)}^4}=\infty$

Use the precise definition of infinite limits to prove the following limits.

$\underset{x\to 0}{\lim }(\frac{1}{x^2}+1)=\infty$

Use the precise definition of infinite limits to prove the following limits.

$\underset{x\to 0}{\lim }(\frac{1}{x^4}-\sin \left(x\right))=\infty$

a. Use a graphing utility to estimate lim x→0 tan 2x / sin x, lim x→0 tan 3x / sin x, and lim x→0 tan 4x / sin x.

Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.

a. With K = 300 and b = 30, what is lim_t→∞ P(t), the carrying capacity of the population?

Horizontal and Vertical Asymptotes

Use limits to determine the equations for all vertical asymptotes.

x² + x ― 6

c. y = ------------------

x² + 2x ― 8

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.

a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.

b. About how fast was the object going when it hit the surface?

[Technology Exercise] 22. Make a table of values for the function at the points x=1.2, x=11/10, x=101/100, x=1001/1000, x=10001/10000, and x = 1.

a. Find the average rate of change of F(x) over the intervals [1,x] for each x≠1 in your table.

b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1.