For the functions in 4653, do the following: (a) Make a table of values of f(x) for x =

Use Figure 1.94 to give approximate values for the following limits (if they exist)

Use Figure 1.95 to estimate the following limits, if they exist. (a) lim x1 f(x) (b) lim x1+ f(x) (c) lim x1 f(x) (d) lim x2 f(x) (e) lim x2+ f(x) (f) lim x2 f(x)

Using Figures 1.96 and 1.97, estimate (a) lim x1 (f(x) + g(x)) (b) lim x1+ (f(x)+2g(x)) (c) lim x1 f(x)g(x) (d) lim x1+ f(x) g(x)

In Exercises 49, draw a possible graph of f(x). Assume f(x) is defined and continuous for all real x

In Exercises 49, draw a possible graph of f(x). Assume f(x) is defined and continuous for all real x

In Exercises 49, draw a possible graph of f(x). Assume f(x) is defined and continuous for all real x

In Exercises 49, draw a possible graph of f(x). Assume f(x) is defined and continuous for all real x

In Exercises 49, draw a possible graph of f(x). Assume f(x) is defined and continuous for all real x

In Exercises 49, draw a possible graph of f(x). Assume f(x) is defined and continuous for all real x

In Exercises 1015, give lim x f(x) and lim x+ f(x)

In Exercises 1015, give lim x f(x) and lim x+ f(x)

In Exercises 1015, give lim x f(x) and lim x+ f(x)

In Exercises 1015, give lim x f(x) and lim x+ f(x)

In Exercises 1015, give lim x f(x) and lim x+ f(x)

In Exercises 1015, give lim x f(x) and lim x+ f(x)

Estimate the limits in Exercises 1617 graphically.

Estimate the limits in Exercises 1617 graphically.

Does f(x) = |x| x have right or left limits at 0? Is f(x) continuous?

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

Use a graph to estimate each of the limits in Exercises 1928. Use radians unless degrees are indicated by .

For the functions in Exercises 2931, use algebra to evaluate the limits lim xa+ f(x), lim xa f(x), and lim xa f(x) if they exist. Sketch a graph to confirm your answers.

For the functions in Exercises 2931, use algebra to evaluate the limits lim xa+ f(x), lim xa f(x), and lim xa f(x) if they exist. Sketch a graph to confirm your answers.

For the functions in Exercises 2931, use algebra to evaluate the limits lim xa+ f(x), lim xa f(x), and lim xa f(x) if they exist. Sketch a graph to confirm your answers.

Estimate how close should be to 0 to make (sin )/ stay within 0.001 of 1.

Write the definition of the following statement both in words and in symbols: lim ha g(h) = K

In Problems 3437, is the function continuous for all x? If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied.

In Problems 3437, is the function continuous for all x? If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied.

In Problems 3437, is the function continuous for all x? If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied.

In Problems 3437, is the function continuous for all x? If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied.

By graphing y = (1 + x) 1/x, estimate lim x0 (1 + x) 1/x. You should recognize the answer you get. What does the limit appear to be?

Investigate lim h0 (1 + h) 1/h numerically.

Investigate lim h0 (1 + h) 1/h numerically.

If p(x)is the function on page 54 giving the price of mailing a first-class letter, explain why limx1 p(x) does not exist.

The notation limx0+ means that we only consider values of x greater than 0. Estimate the limit lim x0+ xx, either by evaluating xx for smaller and smaller positive values of x (say x = 0.1, 0.01, 0.001,...) or by zooming in on the graph of y = xx near x = 0.

In Problems 4345, modify the definition of limit on page 59 to give a definition of each of the following.

In Problems 4345, modify the definition of limit on page 59 to give a definition of each of the following.

In Problems 4345, modify the definition of limit on page 59 to give a definition of each of the following.

For the functions in Problems 4653, do the following: (a) Make a table of values of f(x) for x = 0.1, 0.01, 0.001, 0.0001, 0.1, 0.01, 0.001, and 0.0001. (b) Make a conjecture about the value of lim x0 f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x near 0 such that the difference between your conjectured limit and the value of the function is less than 0.01. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

For the functions in Problems 4653, do the following: (a) Make a table of values of f(x) for x = 0.1, 0.01, 0.001, 0.0001, 0.1, 0.01, 0.001, and 0.0001. (b) Make a conjecture about the value of lim x0 f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x near 0 such that the difference between your conjectured limit and the value of the function is less than 0.01. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

For the functions in Problems 4653, do the following: (a) Make a table of values of f(x) for x = 0.1, 0.01, 0.001, 0.0001, 0.1, 0.01, 0.001, and 0.0001. (b) Make a conjecture about the value of lim x0 f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x near 0 such that the difference between your conjectured limit and the value of the function is less than 0.01. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

For the functions in Problems 4653, do the following: (a) Make a table of values of f(x) for x = 0.1, 0.01, 0.001, 0.0001, 0.1, 0.01, 0.001, and 0.0001. (b) Make a conjecture about the value of lim x0 f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x near 0 such that the difference between your conjectured limit and the value of the function is less than 0.01. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

For the functions in Problems 4653, do the following: (a) Make a table of values of f(x) for x = 0.1, 0.01, 0.001, 0.0001, 0.1, 0.01, 0.001, and 0.0001. (b) Make a conjecture about the value of lim x0 f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x near 0 such that the difference between your conjectured limit and the value of the function is less than 0.01. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

For the functions in Problems 4653, do the following: (a) Make a table of values of f(x) for x = 0.1, 0.01, 0.001, 0.0001, 0.1, 0.01, 0.001, and 0.0001. (b) Make a conjecture about the value of lim x0 f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x near 0 such that the difference between your conjectured limit and the value of the function is less than 0.01. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

For the functions in Problems 4653, do the following: (a) Make a table of values of f(x) for x = 0.1, 0.01, 0.001, 0.0001, 0.1, 0.01, 0.001, and 0.0001. (b) Make a conjecture about the value of lim x0 f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x near 0 such that the difference between your conjectured limit and the value of the function is less than 0.01. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

For the functions in Problems 4653, do the following: (a) Make a table of values of f(x) for x = 0.1, 0.01, 0.001, 0.0001, 0.1, 0.01, 0.001, and 0.0001. (b) Make a conjecture about the value of lim x0 f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x near 0 such that the difference between your conjectured limit and the value of the function is less than 0.01. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

Assuming that limits as x have the properties listed for limits as x c on page 60, use algebraic manipulations to evaluate lim x for the functions in Problems 5463.

In Problems 6471, find a value of the constant k such that the limit exists.

In Problems 6471, find a value of the constant k such that the limit exists.

In Problems 6471, find a value of the constant k such that the limit exists.

In Problems 6471, find a value of the constant k such that the limit exists.

In Problems 6471, find a value of the constant k such that the limit exists.

In Problems 6471, find a value of the constant k such that the limit exists.

In Problems 6471, find a value of the constant k such that the limit exists.

In Problems 6471, find a value of the constant k such that the limit exists.

For each value of  in Problems 7273, find a positive value of such that the graph of the function leaves the window a

For each value of  in Problems 7273, find a positive value of such that the graph of the function leaves the window a

Show that lim x0 (2x + 3) = 3. [Hint: Use Problem 72.]

Consider the function f(x) = sin(1/x). (a) Find a sequence of x-values that approach 0 such that sin(1/x)=0. [Hint: Use the fact that sin() = sin(2) = sin(3) = ... = sin(n)=0.] (b) Find a sequence of x-values that approach 0 such that sin(1/x)=1. [Hint: Use the fact that sin(n/2) = 1 if n = 1, 5, 9,....] (c) Find a sequence of x-values that approach 0 such that sin(1/x) = 1. (d) Explain why your answers to any two of parts (a) (c) show that lim x0 sin(1/x) does not exist.

For the functions in Problems 7677, do the following: (a) Make a table of values of f(x) for x = a+0.1, a+0.01, a + 0.001, a + 0.0001, a 0.1, a 0.01, a 0.001, and a 0.0001. (b) Make a conjecture about the value of lim xa f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x containing a such that the difference between your conjectured limit and the value of the function is less than 0.01 on that interval. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

For the functions in Problems 7677, do the following: (a) Make a table of values of f(x) for x = a+0.1, a+0.01, a + 0.001, a + 0.0001, a 0.1, a 0.01, a 0.001, and a 0.0001. (b) Make a conjecture about the value of lim xa f(x). (c) Graph the function to see if it is consistent with your answers to parts (a) and (b). (d) Find an interval for x containing a such that the difference between your conjectured limit and the value of the function is less than 0.01 on that interval. (In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom of the window.)

This problem suggests a proof of the first property of limits on page 60: lim xc bf(x) = b lim xc f(x). (a) First, prove the property in the case b = 0. (b) Now suppose that b = 0. Let  > 0. Show that if |f(x) L| < /|b|, then |bf(x) bL| < . (c) Finally, prove that if lim xc f(x) = L then lim xc bf(x) = bL. [Hint: Choose so that if |xc| < , then |f(x) L| < /|b|.]

Prove the second property of limits: lim xc (f(x) + g(x)) = lim xc f(x)+ limxc g(x). Assume that the limits on the right exist.

This problem suggests a proof of the third property of limits (assuming the limits on the right exist): lim xc (f(x)g(x)) =  lim xc f(x)  lim xc g(x)

Show f(x) = x is continuous everywhere.

Use Problem 81 to show that for any positive integer n, the function xn is continuous everywhere.

Use Theorem 1.2 on page 60 to explain why if f and g are continuous on an interval, then so are f + g, fg, and f /g (assuming g(x) = 0 on the interval).

In Problems 8486, explain what is wrong with the statement.

In Problems 8486, explain what is wrong with the statement.

In Problems 8486, explain what is wrong with the statement.

In Problems 8788, give an example of:

In Problems 8788, give an example of:

Suppose that limx3 f(x)=7. Are the statements in Problems 8995 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample.

Suppose that limx3 f(x)=7. Are the statements in Problems 8995 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample.

Suppose that limx3 f(x)=7. Are the statements in Problems 8995 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample.

Suppose that limx3 f(x)=7. Are the statements in Problems 8995 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample.

Suppose that limx3 f(x)=7. Are the statements in Problems 8995 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample.

Suppose that limx3 f(x)=7. Are the statements in Problems 8995 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample.

Suppose that limx3 f(x)=7. Are the statements in Problems 8995 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample.

Which of the statements in Problems 96100 are true about every function f(x) such that lim xc f(x) = L? Give a reason for your answer.

Which of the statements in Problems 96100 are true about every function f(x) such that lim xc f(x) = L? Give a reason for your answer.

Which of the statements in Problems 96100 are true about every function f(x) such that lim xc f(x) = L? Give a reason for your answer.

Which of the statements in Problems 96100 are true about every function f(x) such that lim xc f(x) = L? Give a reason for your answer.

Which of the statements in Problems 96100 are true about every function f(x) such that lim xc f(x) = L? Give a reason for your answer.

Which of the statements in Problems 96100 are true about every function f(x) such that lim xc f(x) = L? Give a reason for your answer.