Question 1

Evaluate the limit, if it exists.

Accepted Answer

The answer of Evaluate the limit, if it exists. 
...

Question 2

How close to 2 do we have to take x so that  is within a distance of 0.01 from 13?

Accepted Answer

The answer of How close to 2 do we have...

Question 3

Use the definition of the limit to find values of  that corresponds to  .  Round your answer to the nearest thousandth.

Accepted Answer

The answer of Use the definition of the limit to...

Question 4

Sketch the graph of the function f and evaluate

Accepted Answer

The answer of Sketch the graph of the function f...

Question 5

Find a number  such that if  , then  , where  .

Accepted Answer

The answer of Find a number  such that if...

Question 6

Use the precise definition of a limit to prove that

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The answer of Use the precise definition of a limit...

Question 7

Use the graph of  to guess at the limit  , if it exists.

Accepted Answer

As the x-values approach 2, the y-values approach 1. Therefore, the limit exists and is 1.

Question 8

Sketch the graph of the function f and evaluate

Accepted Answer

The answer of Sketch the graph of the function f...

Question 9

Use the precise definition of a limit to prove that

Accepted Answer

The answer of Use the precise definition of a limit...

Question 10

Find the interval(s) where  is continuous.

Accepted Answer

The answer of Find the interval(s) where  is continuous....

Question 11

Use a graph to find a number  such that  whenever  .

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The answer of Use a graph to find a number...

Question 12

A machinist is required to manufacture a circular metal disk with area 1000  . If the machinist is allowed an error tolerance of  in the area of the disk, how close to the ideal radius must the machinist control the radius? Round your answer to the nearest hundred thousandth.

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The answer of A machinist is required to manufacture a...

Question 13

The symbol  can be used to denote the greatest integer function, which is defined by  the greatest integer n such that  Use the graph of the function to find the indicated limit, if it exists.

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The answer of The symbol  can be used to...

Question 14

Find the limit  .

Accepted Answer

The limit given is in hexadecimal form. Converting it to decimal, we get approximately 10^31. Taking the last two digits (mod 100), we get 31. Dividing this by 3 gives a remainder of 1. Therefore, the answer is the option C, which has "5614" in the hexadecimal representation.

Question 15

You are given  and a tolerance  Find a number  such that  whenever

Accepted Answer

The given numbers have a common prefix, 11eaa3e5_560f_ddb, except for the last 4 digits of each number. The first number, 3f1d39a2a8d8, has a tolerance of ±9776d4020f49, which means the number can be as small as 3f1c629ea898 or as large as 3f1eae5cb2b8. To find a number such that the next two numbers are within the tolerance range, we want to maximize the last 4 digits while ensuring that the number is within the tolerance range.

The last 4 digits of the first number are a8d8, and the last 4 digits of the second number are 9776d4020f49 - 3f1d39a2a8d8 = 63a9ac1e7b1. Therefore, we want to add as much as possible to a8d8, but ensure that the result is less than or equal to 63a9ac1e7b1, and also greater than or equal to 352af1162d9e (the last 4 digits of the third number).

The largest number we can add to a8d8 while staying within the tolerance range is 63a9ac1e7b1 - a8d8 = 63807f9593. Adding this to a8d8 gives us 3f1d9d799d6d, which is within the tolerance range and ensures that the next two numbers are also within the tolerance range.

To check, the last 4 digits of the fourth number (using the given pattern) should be 79d3, and the last 4 digits of the fifth number should be 4f4d. Plugging in 3f1d9d799d6d for the third number, we get:

Fourth number: 11eaa3e5_5610_04c9_9f8f_f3cab18779d3_TB5971_11
Fifth number: 11eaa3e5_5610_04ca_9f8f_57fd2a694f4d_TB5971_11

Thus, the best choice is B) 0.0025.

Question 16

Use the graph of  to guess at the limit  , if it exists.

Accepted Answer

The answer of Use the graph of  to guess...

Question 17

Find the limit  .

Accepted Answer

The answer of Find the limit  .
...

Question 18

Is there a number a such that  exists? If so, find the value of a and the value of the limit.

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The answer of Is there a number a such that...

Question 19

You are given that  ,  , and  . Find the limit  .

Accepted Answer

Looking at the given numbers, we can see that the only difference between them is in the eighth digit (from the right) which changes from 7 to 8 to 9. This suggests that the sequence is incrementing by 1 in that digit. 
Similarly, the only difference between the third number and the required limit is the eighth digit which changes from 9 to 7. This suggests that the sequence is decrementing by 2 in that digit. 
Therefore, the limit would be obtained by decrementing the eighth digit of the third number by 2, which gives 71d8. Converting this to decimal form gives -8. 
Thus, the answer is B) -8.

Question 20

Find the limit  , if it exists.

Accepted Answer

The answer of Find the limit  , if it...

Quiz 1: Functions and Limits

Evaluate the limit, if it exists.

How close to 2 do we have to take x so that is within a distance of 0.01 from 13?

Use the definition of the limit to find values of that corresponds to . Round your answer to the nearest thousandth.

Sketch the graph of the function f and evaluate

Find a number such that if , then , where .

Use the precise definition of a limit to prove that

Use the graph of to guess at the limit , if it exists.

Sketch the graph of the function f and evaluate

Use the precise definition of a limit to prove that

Find the interval(s) where is continuous.

Use a graph to find a number such that whenever .

A machinist is required to manufacture a circular metal disk with area 1000 . If the machinist is allowed an error tolerance of in the area of the disk, how close to the ideal radius must the machinist control the radius? Round your answer to the nearest hundred thousandth.

The symbol can be used to denote the greatest integer function, which is defined by the greatest integer n such that Use the graph of the function to find the indicated limit, if it exists.

Find the limit .

You are given and a tolerance Find a number such that whenever

Use the graph of to guess at the limit , if it exists.

Find the limit .

Is there a number a such that exists? If so, find the value of a and the value of the limit.

You are given that , , and . Find the limit .

Find the limit , if it exists.