Limits

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| < $\epsilon$.

$\underset{x\to 0}{\lim }(x^3+1)=1$

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| < $\epsilon$.

$\underset{x\to 2}{\lim }\frac{1}{x}=\frac{1}{2}; \mathrm{you} \mathrm{may} \mathrm{assume} \delta \le 1$

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| < $\epsilon$.

$\underset{x\to 3}{\lim }\frac{1}{x}=\frac{1}{3}; \mathrm{you} \mathrm{may} \mathrm{assume} \delta \le 1$

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| < $\epsilon$.

$\underset{x\to 3}{\lim }(x^2-2x-3)=0; \mathrm{you} \mathrm{may} \mathrm{assume} \delta \le 1$

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| < $\epsilon$.

$\underset{x\to 1}{\lim }2x^4=2; \mathrm{you} \mathrm{may} \mathrm{assume} \delta \le 1$.

For each limit statement, use algebra to find δ or N in terms of $\epsilon$ or M, according to the appropriate formal limit definition. 

 $\underset{x\to -2^{+}}{lim} (1+\sqrt{x+2})=1$ find δ in terms of $\epsilon$.

For each limit statement in Exercises $41-44$, use algebra to find $\delta$ or $N$ in terms of $\epsilon$ or $M$, according to the appropriate formal limit definition.

$\underset{x\to \infty }{\lim }\frac{x-1}{x}=1$, find $N$ in terms of $\epsilon$.

For each limit statement in Exercises $41-44$, use algebra to find $\delta$ or $N$ in terms of $\epsilon$ or $M$, according to the appropriate formal limit definition.

$\underset{x\to 1^{-}}{\lim }\frac{1}{1-x}=\infty$, find $\delta$ in terms of $M$.

For each limit statement in Exercises $41-44$, use algebra to find $\delta$ or $N$ in terms of $\epsilon$ or $M$, according to the appropriate formal limit definition.

$\underset{x\to \infty }{\lim }\left(x^2+2\right)=\infty$,find $N$ in terms of $M$.

Explain why it makes intuitive sense that $\underset{x\to c}{lim} x=c$ for any real number c. Then use a delta–epsilon argument to prove it. 

Explain why it makes intuitive sense that $\underset{x\to c}{lim} x^2=c^2$ for any real number c. Then use a delta–epsilon argument to prove it. (Hint: You will need to assume that δ ≤ 1 )

Use the preceding two problems and the result of Exercise 22 to calculate the following limit 

$\begin{array}{l}\underset{x\to -1}{lim} x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\cdot \underset{x\to 4}{lim} x\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\underset{x\to \pi }{lim} x\\ \underset{x\to 0}{lim} x^2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\cdot \underset{x\to 5}{lim} x^2{x}_{x\to -2}\\ \underset{x\to 0}{lim} (2x-3)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\cdot \underset{x\to 1}{lim} (1-x)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\cdot \underset{x\to 3}{lim} (3x+1)\end{array}$

When calculating each of these $\underset{x\to c}{lim} f\left(x\right)$, you simply used the value of f(c). Will that method always work for any limit? Why or why not?

Find a formula for the cost $C\left(r\right)$ of producing a gourmet soup can with radius $r$ and height $5$ inches, and answer the following questions:

1.  What is the radius of a can that is $5$ inches tall and costs $30$ cents to produce?
2. Your manager wants you to produce $5$-inch-tall cans that cost between $20$ and $40$ cents. Write this requirement as an absolute value inequality.
3. What range of radii would satisfy your manager? Write an absolute value inequality whose solution set lies inside this range of radii.

Find a formula for the cost $C\left(h\right)$ of producing a gourmet soup can with height $h$ and radius $2$ inches, and answer the following questions:

1. What is the height of a can that has radius $2$ inches and costs $45$ cents to produce?
2. Your manager wants you to produce $2$-inch-radius cans that cost between $40$ and $50$ cents. Write this requirement as an absolute value inequality.
3. What range of heights would satisfy your manager? Write an absolute value inequality whose solution set lies within this range of heights.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$in Exercises $47-60$.

$\underset{x\to 1}{\lim }\left(2x+4\right)=6$.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to 2}{\lim }\left(3-4x\right)=-5$.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

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

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to -3}{\lim }\left(1-x\right)=4$.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to 4}{\lim }\left(6x-1\right)=23$.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to 8}{\lim }\left(3x-11\right)=13$.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to 0}{\lim }\left(3x^2+1\right)=1$.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to 3}{\lim }\left(x^2-6x+11\right)=2$.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to 1}{\lim }\left(2x^2-4x+3\right)=1$.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47-60$.

$\underset{x\to 2}{\lim }\left(3x^2-12x+15\right)=3$.

Write delta-epsilon proofs for each of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L$ in Exercises $47–60.$

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

Write delta-epsilon proofs for each of the limit statements  in Exercises $47–60.$

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

Write delta-epsilon proofs for each of the limit statements  in Exercises $47–60.$

$\underset{x\to 5^{+}}{\lim }\sqrt{x-5}=0$

Write delta-epsilon proofs for each of the limit statements  in Exercises $47–60.$

$\underset{x\to 2^{+}}{\lim }3\sqrt{2x-4}=0$

For each of the limit statements in Exercises 61-66, write a $\delta -M,N-ϵ$, or $N-M$ proof, according to the type of limit statement.

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

For each of the limit statements in Exercises 61-66, write a $\delta -M,N-ϵ$, or $N-M$ proof, according to the type of limit statement.
$\underset{x\to -2^{-}}{\lim }\frac{1}{x+2}=-\infty$

For each of the limit statements in Exercises 61-66, write a $\delta -M,N-ϵ$, or $N-M$ proof, according to the type of limit statement.
$\underset{x\to \infty }{\lim }\frac{2x-1}{x}=2$

For each of the limit statements in Exercises 61-66, write a $\delta -M,N-ϵ$, or $N-M$ proof, according to the type of limit statement.
$\underset{x\to -\infty }{\lim }\frac{2x-1}{x}=2$

For each of the limit statements in Exercises 61-66, write a $\delta -M,N-ϵ$, or $N-M$ proof, according to the type of limit statement.
$\underset{x\to \infty }{\lim }(3x-5)=\infty$

For each of the limit statements in Exercises 61-66, write a $\delta -M,N-ϵ$, or $N-M$ proof, according to the type of limit statement.
$\underset{x\to -\infty }{\lim }(3x-5)=-\infty$

Prove each of the limit statements in Exercises 67–72. You

will have to bound $\delta$.

$\underset{x\to 3}{\lim }\left(x^2-2x-3\right)=0$

Prove each of the limit statements in Exercises 67–72. You will have to bound $\delta$.

$\underset{x\to -1}{\lim }\left(x^2-2x-3\right)=0$

Prove each of the limit statements in Exercises 67–72. You

will have to bound $\delta$.

$\underset{x\to 5}{\lim }\left(x^2-6x+7\right)=2$

Prove each of the limit statements in Exercises 67–72. You

will have to bound $\delta$.

$\underset{x\to 1}{\lim }\left(x^2-6x+7\right)=2$

Prove each of the limit statements in Exercises 67–72. You

will have to bound $\delta$.

$\underset{x\to 2}{\lim }\frac{4}{x^2}=1$

For each of the limit statements in Exercises 61-66, write a $\delta -M,N-ϵ$, or $N-M$ proof, according to the type of limit statement.

$\underset{x\to 3}{\lim }\frac{18}{x^2}=2$

Prove each of the limit statements in Exercises 67–72. You
will have to bound $\delta$.

$\underset{x\to 3}{\lim } \frac{18}{x^2} =2$ 

Problem Zero: 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 f is both left and right continuous at x = c, then f is continuous at x = c. 

(b) True or False: If f is continuous on the open interval (0, 5), then f is continuous at every point in (0, 5).

(c) True or False: If f is continuous on the closed interval [0, 5], then f is continuous at every point in [0, 5].

(d) True or False: If f is continuous on the interval (2, 4), then f must have a maximum value and a minimum value on (2, 4).

(e) True or False: If f (3) = −5 and f (9) = −2, then there must be a value c at which f (c) = −3.

(f) True or False: If f is continuous everywhere, and if f (−2) = 3 and f (1) = 2, then f (x) must have a root somewhere in (−2, 1).

(g) True or False: If f is continuous everywhere, and if f (0) = −2 and f (4) = 3, then f (x) must have a root somewhere in (0, 4).

(h) True or False: If f (0) = f (6) = 0 and f (2) > 0, then f (x) is positive on the entire interval (0, 6).

Finding roots of piecewise-defined functions: For each function f that follows, find all values x = c for which f(c) = 0. Check your answers by sketching a graph of f. 

$\begin{array}{r}f\left(x\right)=\left\{\begin{array}{rl}4-x^2,& \text{ if }x<0\\ x+1,& \text{ if }x\ge 0\end{array}\right.\\ f\left(x\right)=\left\{\begin{array}{rl}x+1,& \text{ if }x<0\\ 4-x^2,& \text{ if }x\ge 0\end{array}\right.\\ f\left(x\right)=\left\{\begin{array}{rl}2x-1,& \text{ if }x\le 1\\ 2x^2+x-3,& \text{ if }x>1\end{array}\right.\end{array}$

Interesting trigonometric limits: For each of the functions that follow, use a calculator or other graphing utility to examine the graph of f near x = 0. Does it appear that f is continuous at x = 0? Make sure your calculator is set to radian mode 

$f\left(x\right)=\left\{\begin{array}{r}\frac{1}{x}\sin \left(x\right),\text{ if }x\ne 0\\ 1,\text{ if }x=0\end{array}\right.$

1. 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 f is both left and right continuous at x=c, then f is continuous at x=c.

(b) True or False: If f is continuous on the open interval (0,5), then f is continuous at every point in (0,5).

(c) True or False: If f is continuous on the closed interval [0,5], then f is continuous at every point in [0,5].

(d) True or False: If f is continuous on the interval (2,4), then f must have a maximum value and a minimum value on (2,4).

(e) True or False: If f(3)=-5 and f(9)=-2, then there must be a value c at which f(c)=-3.

(f) True or False: If f is continuous everywhere, and if f(-2)=3 and f(1)=2, then f(x) must have a root somewhere in (-2,1)

(g) True or False: If f is continuous everywhere, and if f(0)=-2 and f(4)=3, then f(x) must have a root somewhere in (0,4).

(h) True or False: If f(0)=f(6)=0 and f(2)>0, then f(x) is positive on the entire interval (0,6)

Logical existence statements: Determine whether each of the statements that follow is true or false. Justify your answers. 

If x is an integer, then there exists some positive integer y such that |y| = x. 

If x is a positive integer, then there exists some negative integer y such that |y| = x.  

If x ∈ [−2, 2], then there exists some y ∈ (0, 4) such that y = x 2. 

If x ∈ [0, 100], then there exists some y ∈ [−10, 10] such that x = y2. 

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

(a) The graph of a function with $f\left(4\right)=2$ that has a removable discontinuity at x=4.

(b) The graph of a function that is continuous on its domain but not continuous at x=0.

(c) The graph of a function that is continuous on (0,2] and (2,3) but not on (0,3).

If $f$ is a continuous function, what can you say about $\underset{x\to 1}{\lim }f\left(x\right)?$