Question 1

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

Accepted Answer

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

Question 2

Suppose $$\lim_{x 	o 2} f(x) = 5. $$According to the epsilon-delta definition, what does this mean?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists a $$\delta \u003e 0$$ such that $$|f(x) - 5| \u003c \epsilon$$ whenever $$0 \u003c |x - 2| \u003c \delta.$$

Question 3

Given $$f(x) = 2x - 5$$, find the value of $$\delta$$ such that $$|f(x) - (-1)| \u003c 0.5$$ whenever $$|x - 2| \u003c \delta.$$

Accepted Answer

$$\delta = 0.25$$

Question 4

If $$f(x) = x^2$ and $$\lim_{x 	o 1} f(x) = 1$$, which of the following is the correct $$\delta$$ for $$\epsilon = 0.1$$?

Accepted Answer

$$\delta = 0.316$$

Question 5

Which statement best explains the role of $$\epsilon in $$the epsilon-delta definition of a limit?

Accepted Answer

$$\epsilon$$ represents how close $$f(x)$$ must be to $$L.$$

Question 6

In the context of the epsilon-delta definition, what does the interval $$a - \delta \u003c x \u003c a + \delta$$ represent?

Accepted Answer

The set of $$x-$$values within $$\delta$$ units of $$a$$, excluding $$a$$ itself.

Question 7

Suppose $$f(x) = x^2$ and you want $$|f(x) - 4| \u003c 0.5$$ for x$ near 2$. What is the largest $$\delta$$ you can choose?

Accepted Answer

$$\delta = 0.707$$

Question 8

Which of the following is a correct interpretation of the phrase: 'for every $$\epsilon \u003e 0$$, there exists a $$\delta \u003e 0$$'?

Accepted Answer

No matter how small a tolerance you set for $$f(x)$$, you can find a range around $$a$$ where $$f(x)$$ stays within that tolerance.

Question 9

Given $$f(x) = x^2$, a = 1$, and $$\epsilon = 0.1$$, which inequality must $$\delta$$ satisfy to ensure $$|f(x) - 1| \u003c 0.1$$ whenever $$|x - 1| \u003c \delta$$?

Accepted Answer

$$|x - 1| \u003c \delta$$ where $$\delta \u003c 0.316$$

Question 10

If $$f(x) = 2x - 5$$ and $$a = 2$$, what is the value of $$f(a)$$ and what does this represent in the context of limits?

Accepted Answer

$$f(2) = -1$$, which is the value the function approaches as $$x$$ approaches $$2.$$

Question 11

A chemist is measuring the temperature of a reaction, modeled by $$f(x) = 2x - 5$$, where $$x is $$time in minutes. If the desired temperature is $$-1$$ degrees, how close must the time $$x be to$$ $$2$$ minutes to ensure the temperature is within $$0.2$$ degrees of the target?

Accepted Answer

$$|x - 2| \u003c 0.1$$

Question 12

Which of the following is NOT a requirement for the epsilon-delta definition of a limit?

Accepted Answer

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

Question 13

Suppose $$f(x) = x^2$ and you want $$|f(x) - 9| \u003c 1$$ for x$ near 3$. What is the largest $$\delta$$ you can choose?

Accepted Answer

$$\delta = 1$$

Question 14

A medical device is calibrated so that its output $$f(x)$$ must be within $$0.1$$ units of the target value $$L$$ when the input $$x is $$within $$\delta$$ units of $$a. If$$ $$f(x) = 2x - 5$$, $$a = 2$$, and $$L = -1$$, what is the value of $$\delta$$ for $$\epsilon = 0.1$$?

Accepted Answer

$$\delta = 0.05$$

Precise Definitions of Limits (Epsilon-Delta Definition)

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