Limits

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N  that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.   

$\underset{x\to -\infty }{lim}{3}^x=0, \epsilon =\frac{1}{4}, \mathrm{find} \mathrm{smallest}-\mathrm{magnitude} N<0$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N  that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.    

$\underset{x\to \infty }{lim}(4-x^2)=-\infty , M=-100, \mathrm{find} \mathrm{smallest} N>0$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N  that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.     

$\underset{x\to \infty }{lim}(4-x^2)=-\infty , M=-10000, \mathrm{find} \mathrm{smallest} N>0$

Every month, Jack hides \(50 under a broken floorboard to save up for a new boat. After t months of saving, he will have $F\left(t\right)=50t$ dollars.

(a) The boat Jack wants costs at least \)7,465. How many months does Jack have to save money before he will have enough to pay for the boat? Illustrate this information on a graph of F(t). 

(b) Suppose a different boat costs M dollars. Will there be a value t equal to N for which $F\left(N\right)>M$? What does this mean in real-world terms? Illustrate the roles of M and N on a graph of F(t).

 Len’s company produces different-sized cylindrical cans that are each 6 inches tall. The cost to produce a can with radius r is $C\left(r\right)=10{\mathrm{\pi r}}^2+24\mathrm{\pi r}$cents.

(a) Len’s boss wants him to construct the cans so that the cost of each can is within 25 cents of \(4.00. Given these cost requirements, what is the acceptable range of values for r? 

(b) Len’s boss now says that he wants the cans to cost within 10 cents of \)4.00. Under these new cost requirements, what is the acceptable range of values for r? 

(c) Interpret this problem in terms of delta and epsilon ranges. Specifically, what is c? What is L? What is epsilon for part (a) and part (b)? What are the corresponding values of delta? Illustrate these values of c, L, epsilon, and delta on a graph of C(r).

You work for a company that sells velvet Elvis paintings. The function $N\left(p\right)=9.2p^2-725p+16333$ predicts the number N of velvet Elvis paintings that your company will sell if they are priced at p dollars each, and is shown in the following graph. The Presley estate does not allow you to charge more than $50 per painting.

(a) Use a graphing utility to estimate the price your company should charge per painting if it wishes to sell 6000 velvet Elvis paintings. 

(b) Find the range of prices that would enable your company to sell between 5000 and 7000 velvet Elvis paintings. 

(c) Interpret this problem in terms of delta and epsilon ranges. Specifically, what is c? What is L? What is epsilon? What is the corresponding value of delta? Illustrate these values of c, L, epsilon, and delta on a graph of N(p).

$$\begin{gathered} \mathrm{Prove} \mathrm{that} \underset{\mathrm{x}\to 1}{\lim 3}\mathrm{x}=3, \mathrm{with} \mathrm{these} \mathrm{steps}: \\ \left(\mathrm{a}\right) \mathrm{What} \mathrm{is} \mathrm{the} \delta –\epsilon \mathrm{statement} \mathrm{that} \mathrm{must} \mathrm{be} \mathrm{shown} \mathrm{to} \mathrm{prove} \mathrm{that} \underset{\mathrm{x}\to 1}{\lim 3}\mathrm{x}=3? \\ \left(\mathrm{b}\right) \mathrm{Argue} \mathrm{that} x \in (1 - \delta , 1) \cup (1, 1 + \delta ) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} -\delta < x - 1 < \delta , \\ with x \ne 1. \mathrm{Then} \mathrm{use} \mathrm{algebra} \mathrm{to} \mathrm{show} \mathrm{that} \mathrm{this} \mathrm{means} \mathrm{that} 0 < |x - 1| < \delta . \\ \left(\mathrm{c}\right) \mathrm{Argue} \mathrm{that} 3x \in (3 - \epsilon , 3 + \epsilon ) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} -\epsilon < 3(x - 1) < \epsilon . \\ \mathrm{Then} \mathrm{use} \mathrm{algebra} \mathrm{to} \mathrm{show} \mathrm{that} \mathrm{this} \mathrm{means} \mathrm{that} 3|x - 1| < \epsilon . \\ \left(\mathrm{d}\right) \mathrm{Given} \mathrm{any} \mathrm{particular} \mathrm{\epsilon } > 0, \mathrm{what} \mathrm{value} \mathrm{of} \mathrm{\delta } \mathrm{would} \mathrm{guarantee} \mathrm{that} \\ \mathrm{if} 0 < |x - 1| < \delta , \mathrm{then} 3|x - 1| < \epsilon ? \mathrm{Your} \mathrm{answer} \mathrm{will} \mathrm{depend} \mathrm{on} \epsilon . \\ \left(\mathrm{e}\right) \mathrm{Put} \mathrm{the} \mathrm{previous} \mathrm{four} \mathrm{parts} \mathrm{together} \mathrm{to} \mathrm{prove} \mathrm{the} \mathrm{limit} \mathrm{statement}. \end{gathered}$$

Prove that $\underset{x\to 2}{\lim }(7-x)=5$, with these steps:

(a) What is the $\delta -\epsilon$ statement that must be shown to prove that $\underset{x\to 2}{\lim }(7-x)=5$?

(b) Argue that $x\in (2-\delta , 2) \cup (2, 2+\delta )$ if and only if$-\delta <x-2<\delta$. Then use algebra to show that this means that $0<\left|x-2\right|<\delta$.

(c) Argue that $7-x\in (5-\epsilon , 5+\epsilon )$ if and only if $-\epsilon <2-x<\epsilon$. Then use algebra to show that this means that $\left|x-2\right|<\epsilon$.

(d) Given any particular $\epsilon >0$, what value of $\delta$ would guarantee that if $0<\left|x-2\right|<\delta$, then $\left|x-2\right|<\epsilon$? Your answer will depend on $\epsilon$.

(e) Put the previous four parts together to prove the limit statement.

$$\begin{gathered} \mathrm{Prove} \mathrm{that} \underset{\mathrm{x}\to 0^{+}}{\lim }\frac{1}{\mathrm{x}}=\infty , \mathrm{with} \mathrm{these} \mathrm{steps}: \\ \left(\mathrm{a}\right) \mathrm{What} \mathrm{is} \mathrm{the} M–\delta \mathrm{statement} \mathrm{that} \mathrm{must} \mathrm{be} \mathrm{shown} \mathrm{to} \mathrm{prove} \mathrm{that}\underset{\mathrm{x}\to 0^{+}}{\lim }\frac{1}{\mathrm{x}}=\infty ? \\ \left(\mathrm{b}\right) \mathrm{Argue} \mathrm{that} x \in (0, 0 + \delta ) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} 0 < x < \delta . \\ \left(\mathrm{c}\right) \mathrm{Argue} \mathrm{that} \frac{1}{\mathrm{x}} \in (M,\infty ) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} x < \frac{1}{M} . \mathrm{You} \mathrm{may} \mathrm{assume} \mathrm{that} M > 0. \\ \left(\mathrm{d}\right) \mathrm{Given} \mathrm{any} \mathrm{particular} M > 0, \mathrm{what} \mathrm{value} \mathrm{of} \delta \mathrm{would} \mathrm{guarantee} \mathrm{that} \\ \mathrm{if} 0 < x < \delta , \mathrm{then} x < \frac{1}{M}? \mathrm{Your} \mathrm{answer} \mathrm{will} \mathrm{depend} \mathrm{on} M. \\ \left(\mathrm{e}\right) \mathrm{Put} \mathrm{the} \mathrm{previous} \mathrm{four} \mathrm{parts} \mathrm{together} \mathrm{to} \mathrm{prove} \mathrm{the} \mathrm{limit} \mathrm{statement}. \end{gathered}$$

$$\begin{gathered} \mathrm{Prove} \mathrm{that} \underset{\mathrm{x}\to \infty }{\lim }\frac{1}{\mathrm{x}}=0, \mathrm{with} \mathrm{these} \mathrm{steps}: \\ \left(\mathrm{a}\right) \mathrm{What} \mathrm{is} \mathrm{the} \epsilon –N \mathrm{statement} \mathrm{that} \mathrm{must} \mathrm{be} \mathrm{shown} \mathrm{to} \mathrm{prove} \mathrm{that} \underset{\mathrm{x}\to \infty }{\lim }\frac{1}{\mathrm{x}}=0? \\ \left(\mathrm{b}\right) \mathrm{Argue} \mathrm{that} x \in (N,\infty ) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if} x > N. \\ \left(\mathrm{c}\right) \mathrm{Argue} \mathrm{that} \frac{1}{\mathrm{x}} \in (0-\epsilon , 0+\epsilon ) \mathrm{if} \mathrm{and} \mathrm{only} \mathrm{if}-\epsilon < \frac{1 }{x}< \epsilon . \\ \mathrm{Then} \mathrm{argue} \mathrm{that} \mathrm{for} \mathrm{this} \mathrm{limit}, \mathrm{it} \mathrm{suffices} \mathrm{to} \mathrm{consider} 0 < \frac{1}{x} < \epsilon . \\ \left(\mathrm{d}\right) \mathrm{Given} \mathrm{any} \mathrm{particular} \epsilon , \mathrm{what} \mathrm{value} \mathrm{of} N > 0 \mathrm{would} \mathrm{guarantee} \mathrm{that} \\ \mathrm{if} x > N, \mathrm{then} 0 < \frac{1 }{x}< \epsilon ? \mathrm{Your} \mathrm{answer} \mathrm{will} \mathrm{depend} \mathrm{on} \epsilon . \\ \left(\mathrm{e}\right) \mathrm{Put} \mathrm{the} \mathrm{previous} \mathrm{four} \mathrm{parts} \mathrm{together} \mathrm{to} \mathrm{prove} \mathrm{the} \mathrm{limit} \mathrm{statement}. \end{gathered}$$

As you have already seen, sometimes$\underset{x\to c}{lim} f\left(x\right)$ is equal to f(c), and sometimes it is not. As we will see in Section 1.4, when the limit of a function f as x → c does happen to be equal to the value of f(x) at x = c, we say that the function f is continuous at x = c.

(a)   You may have heard the following loose, only partially accurate definition of continuity in a previous class: A function is continuous if you “can draw it without picking up your pencil.” Why does it make sense that this would be related to the definition just presented of continuity in terms of limits? 

(b)   State the limit-definition of continuity with a formal δ–$\in$ statement .

(c)   The function $f\left(x\right)=x^2$ is continuous at every point. Use this fact and the formal definition of continuity to calculate $\underset{x\to 2}{lim} x^2,\underset{x\to 5}{lim} x^2,\text{ and }\underset{x\to -4}{lim} x^2$ .

State the limit-definition of continuity with a formal $\delta -\epsilon$statement.

Read the section and make your own summary of the material.

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: If $x\ne c$, then $\left|x-c\right|$ is strictly greater than zero.

(b) True or False: If $\left|x-c\right|$ is strictly greater than zero, then $x\ne c$.

(c) True or False: x is a solution of $0<\left|x-c\right|<\delta$ if and only if $c-\delta <x<c+\delta$.

(d) True or False: If $0<\left|x-c\right|<\delta$, then $x\in \left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$.

(e) True or False: If $\left|f\left(x\right)-L\right|<\in$ then $L-\in <f\left(x\right)<L+\in$.

(f) True or False: If $f\left(x\right)\in \left(L-\in ,L+\in \right)$ then $0<f\left(x\right)<\left|L+\in \right|$.

(g) True or False: The fact that $0<\left|x-3\right|<0.25$ guarantees that $\left|\left(2x-1\right)-5\right|<0.5$ proves that 

(h) True or False: $\underset{x\to 3}{\lim }(2x-1)=5$ means that for all $\delta >0$ there is some $\epsilon >0$ such that if $0<\left|x-c\right|<\delta$, then $\left|(2x-1)-5\right|<\epsilon$.

Find the solution sets of each of the following inequalities.

$0 < |x - 2| < 0.5$

Find the solution sets of each of the following inequalities.

$0 < |x + 5| < 0.1$

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. 

(a) A function f with values given in the following table but whose limit as x → 2 is not equal to 5:

(b) An inequality involving absolute values whose solution set is $(2.75,3)\cup (3,3.25)$

(c) An inequality involving absolute values whose solution set is $(-0.01,0)\cup (0,0.01)$

Write down the formal delta-epsilon statement you would have to prove in order to prove the limit statement. 

$\underset{x\to -2}{lim}\frac{3}{x+1}=-3$

Suppose you show that $\left|\right(1-2x)-(-5\left)\right|<0.05$ for all x with $0<|x-3|<0.025$Explain why this does not prove that  $\underset{x\to 3}{lim}(1-2x)=-5$

Write down a mathematical equation that expresses the sentence “x is not equal to 5, and the distance between x and 5 is less than $0.01$.” Then write an equation that means “the distance between $f\left(x\right) \mathrm{and} -2$ is less than $0.5$.” 

Why do we have $0<|x-c|<\delta$ instead of just $|x-c|<\delta$ in Definition 1.10? 

Write each of the inequalities in interval notation:

$0<|x-2|<0.1$

Write each of the inequalities in interval notation:

$0<|x+3|<0.05$

Write each of the inequalities in interval notation:

$\left|\left(x^2-1\right)+3\right|<0.5$

Write each of the inequalities in interval notation:

$\left|\left(3x+1\right)-2\right|<0.1$

Write each of the inequalities in interval notation:

$\left|f\left(x\right)-L\right|<\in$

Write each of the following inequalities in interval notation:

0 < |x − c| < δ

Determine whether each implication that follows is true or

false. Use graphs to justify any implications that are true and

counterexamples for any implications that are false.

If $0<|x-2|<1,$then $\left|x^2-4\right|<0.5.$

Determine whether each implication that follows is true or

false. Use graphs to justify any implications that are true and

counterexamples for any implications that are false.

If $0<|x-2|<0.2,$then $\left|x^2-4\right|<1.$

Determine whether each implication that follows is true or

false. Use graphs to justify any implications that are true and

counterexamples for any implications that are false.

If $0<|x-0|<0.75,$then $\left|x^2-0\right|<0.5.$

Determine whether each implication that follows is true or

false. Use graphs to justify any implications that are true and

counterexamples for any implications that are false.

If 0$0<|x+2|<0.1,$then $\left|x^2-4\right|<0.4.$

Determine whether the implication that follows is true or false. Use graphs to justify any implications that are true, and counter examples for any implications that are false.

If $0<\left|x+2\right|<0.075$, then $\left|x^2-4\right|<0.4$.

In Example 2 we proved that $\underset{x\to 2}{\lim }\left(x^2-4x+5\right)=1$. Use the proof to find values of $\delta$ corresponding to

Part (a):$\in =1$

Part (b):$\in =0.1$

Part (c):$\in =0.01$

Illustrate that your choices of $\delta$ work by examining a graph of $f\left(x\right)=x^2-4x+5$and sketching appropriate $\in$ and $\delta$ intervals.

In Example 4 we proved that $\underset{x\to 2}{\lim }5x^4=80$. Use the proof to find values of $\delta$ corresponding to

Part (a):$\in =5$

Part (b):$\in =0.01$

Part (c):$\in =350$

Illustrate that your choices of $\delta$ work by examining a graph of $f\left(x\right)=5x^4$and sketching appropriate $\in$ and $\delta$ intervals.

Use algebra to solve the inequality $0<\left|x-c\right|<\delta$ and show that its solution set is $x\in \left(c-\delta ,c\right)\cup \left(c,c+\delta \right)$.

Use algebra to solve the inequality $\left|f\left(x\right)-L\right|<\in$and show that its solution set is $f\left(x\right)\in \left(L-\in ,L+\in \right)$.

Suppose $f\left(x\right)=mx+b$is a linear function with $m\ne 0$, and let c be any real member.

Part (a): Show that for all $\in >0$, if $0<\left|x-c\right|<\frac{\in }{\left|m\right|}$, then $\left|f\left(x\right)-f\left(c\right)\right|<\in$.

Part (b): What does the implication in part (a) have to do with limits?

Part (c): Illustrate the implication in part (a) with a labeled graph. Explain in terms of slopes why it makes sense that given $\in >0$, the corresponding $\delta >0$ is $\delta =\frac{\in }{\left|m\right|}$.

Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication is true. Then verify and support your answers with labeled graphs.

If $0<\left|x-2\right|<\delta$, then $\left|\left(3x-1\right)-5\right|<0.25$.

Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication is true. Then verify and support your answers with labeled graphs.

If $0<\left|x-3\right|<\delta$, then $\left|\frac{1}{x}-\frac{1}{3}\right|<0.2$

Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication is true. Then verify and support your answers with labeled graphs.

If $x\in \left(1,1+\delta \right)$, then $\left|\sqrt{x-1}-0\right|<0.5$

Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication is true. Then verify and support your answers with labeled graphs.

If $x\in \left(3-\delta ,3\right)$, then $\frac{1}{3-x}>1000$.

Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.

$\text{ If }x>N\text{, then }\left|\frac{1}{x^2}-0\right|<0.001$

Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.
$\text{ If }x>N\text{, then }1-2x<-500$

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon$.

$\underset{x\to 3}{\lim }(x+5)=8$

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon$.

$\underset{x\to -2}{\lim }(4-2x)=8$

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon$.

$\underset{x\to 0}{\lim }(3-4x)=3$

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon$.

$\underset{x\to 1}{\lim }(3x+8)=11$

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon$.

$\underset{x\to 0}{\lim }(5x^2-1)=-1$

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon$.

$\underset{x\to 3}{\lim }(x^2-6x+5)=-4$

For each limit statement , use algebra to find δ > 0 in terms of $\epsilon$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $\epsilon$.

$\underset{x\to 2}{\lim }(x^2-4x+6)=2$