91Ó°ÊÓ

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of $f\left(x\right)$as $x→c$is the value $f\left(c\right)$.

(c) The limit of $f\left(x\right)$as $x→c$might exist even if the value of $f\left(c\right)$does not.

(d) The two-sided limit of $f\left(x\right)$as $x→c$exists if and only if the left and right limits of $f\left(x\right)$exists as $x→c$.

(e) If the graph of $f$has a vertical asymptote at $x=5$, then $\underset{x→5}{\lim }f\left(x\right)=∞$.

(f) If $\underset{x→5}{\lim }f\left(x\right)=∞$, then the graph of $f$has a vertical asymptote at $x=5$.

(g) If $\underset{x→2}{\lim }f\left(x\right)=∞$, then the graph of $f$has a horizontal asymptote at $x=2$.

(h) If$\underset{x→∞}{\lim }f\left(x\right)=2$, then the graph of$f$has a horizontal asymptote at$y=2$.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: For $\underset{x→c}{\lim }f\left(x\right)$ to be defined, the function f must be defined at x = c.

(b) True or False: We can calculate a limit of the form $\underset{x→c}{\lim }f\left(x\right)$ simply by finding f(c).

(c) True or False: If $\underset{x→c}{\lim }f\left(x\right)=10,$ then f(c) = 10.

(d) True or False: If f(c) = 10, then $\underset{x→c}{\lim }f\left(x\right)=10.$

(e) True or False: A function can approach more than one limit as x approaches c.

(f) True or False: If $\underset{x→4}{\lim }f\left(x\right)=10,$ then we can make f(x) as close to 4 as we like by choosing values of x sufficiently close to 10.

(g) True or False: If $\underset{x→6}{\lim }f\left(x\right)=∞,$ then we can make f(x) as large as we like by choosing values of x sufficiently close to 6.

(h) True or False: If $\underset{x→∞}{\lim }f\left(x\right)=100,$ then we can find values of f(x) between 99.9 and 100.1 by choosing values of x that are sufficiently large.

Sketch a labeled graph of a function that fails to satisfy the hypothesis of the Intermediate Value Theorem, and illustrate on your graph that the conclusion of the Intermediate Value Theorem does not necessarily hold.

Describe the punctured interval around $x=2$that has a radius of $3$ and the punctured interval around$x=4$ that has a radius of $0.25$.

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere.

$f\left(x\right)=x^4−3x^2−2,[a,b]=[−2,2]$

Use the Extreme Value Theorem to show that each function f has both a maximum and a minimum value on [a, b]. Then use a graphing utility to approximate values M and m in [a, b] at which f has a maximum and a minimum, respectively. You may assume that these functions are continuous everywhere.

$f\left(x\right)=3−2x^2+x^3,[a,b]=[−1,2]$