Question 1

Which of the following best describes the precise definition of the limit $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

For any number $$\epsilon \u003e 0$$, there is a corresponding number $$\delta \u003e 0$$ such that $$|f(x) - L| \u003c \epsilon$$ whenever $$0 \u003c |x - a| \u003c \delta.$$

Question 2

Given $$f(x) = 2x + 3$$, what is the largest value of $$\delta$$ such that $$|f(x) - 5| \u003c 0.01$$ whenever $$|x - 1| \u003c \delta$$?

Accepted Answer

$$\delta = 0.005$$

Question 3

If $$\lim_{x 	o a} f(x) = L$$, which of the following must be true?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that $$|f(x) - L| \u003c \epsilon$$ whenever $$0 \u003c |x - a| \u003c \delta.$$

Question 4

Suppose you want to prove $$\lim_{x 	o 1} (2x + 3) = 5$$ using the precise definition of a limit. What is the correct first step?

Accepted Answer

Let $$\epsilon \u003e 0$$ and find $$\delta$$ such that $$|2x + 3 - 5| \u003c \epsilon$$ whenever $$|x - 1| \u003c \delta.$$

Question 5

For the function $$f(x) = 2x + 3$$, what is the value of $$x$$ when $$f(x) = 5.01$$?

Accepted Answer

$$x = 1.005$$

Question 6

Which step is NOT part of the process for proving $$\lim_{x 	o a} f(x) = L$$ using the precise definition?

Accepted Answer

Assume $$\delta is $$arbitrary and find $$\epsilon in $$terms of $$\delta.$$

Question 7

If $$f(x) = 2x + 3$$, what is the value of $$x$$ when $$f(x) = 4.99$$?

Accepted Answer

$$x = 0.995$$

Question 8

In the context of the precise definition of a limit, what does $$\delta$$ represent?

Accepted Answer

The maximum distance $$x$$ can be from $$a so $$that $$f(x) is $$within $$\epsilon of$$ $$L.$$

Question 9

If you are given $$|f(x) - L| \u003c \epsilon$$ and $$f(x) = 2x + 3$$, $$L = 5$$, $$a = 1$$, what inequality must $$x$$ satisfy?

Accepted Answer

$$|x - 1| \u003c \frac{\epsilon}{2}$$

Question 10

A medical device is calibrated so that its output $$f(x)$$ approaches a target value $$L as $$the input $$x$$ approaches $$a. If $$the device must be within $$0.01$$ units of $$L$$ when $$x is $$within $$\delta$$ units of $$a$$, what is the role of $$\delta in $$this scenario?

Accepted Answer

$$\delta is $$the tolerance for the input $$x to $$ensure the output is within $$0.01$$ units of $$L.$$

Question 11

Which of the following statements is TRUE regarding the steps for proving a limit using the precise definition?

Accepted Answer

You must show that for every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that $$|f(x) - L| \u003c \epsilon$$ whenever $$0 \u003c |x - a| \u003c \delta.$$

Question 12

A chemical reaction's rate $$f(x) is $$modeled by $$f(x) = 2x + 3. If $$the rate must be within $$0.01 of 5 $$when the concentration $$x is $$near 1, what is the largest interval around $$x = 1$$ that guarantees this accuracy?

Accepted Answer

$$|x - 1| \u003c 0.005$$

Question 13

If $$f(x) = 2x + 3$$ and you want $$|f(x) - 5| \u003c \epsilon$$, what should $$\delta be in $$terms of $$\epsilon$$?

Accepted Answer

$$\delta = \frac{\epsilon}{2}$$

Question 14

Which of the following is a correct application of the precise definition of a limit in a real-world scenario?

Accepted Answer

Ensuring a medication dosage $$f(x) is $$within $$\epsilon$$ units of the target dose $$L$$ whenever the measured input $$x is $$within $$\delta$$ units of the ideal input $$a.$$

Question 15

If $$f(x) = 2x + 3$$ and $$a = 1$$, what is the value of $$f(x)$$ when $$x = 1.005$$?

Accepted Answer

$$f(x) = 5.01$$

Question 16

Which of the following is NOT a requirement for the precise definition of a limit?

Accepted Answer

The function $$f(x)$$ must be defined at $$x = a.$$

2.7 Precise Definition of Limits in Calculus

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