Q: If $$\lim\limits_{x \to a^-} f(x) \neq \lim\limits_{x \to a^+} f(x)$$, what can be concluded?

The limit $$\lim\limits_{x \to a} f(x)$$ does not exist.

Q: Suppose $$f(x)$$ models the temperature in a chemical reaction over time. If $$\lim\limits_{x \to 10^-} f(x) = 100$$ and $$\lim\limits_{x \to 10^+} f(x) = 120$$, what can you conclude?

The limit $$\lim\limits_{x \to 10} f(x)$$ does not exist.

Question 1

Which of the following best describes the meaning of $$\lim\limits_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ approaches $$L.$$

Question 2

What does $$\lim\limits_{x 	o a^-} f(x)$$ represent?

Accepted Answer

The limit of $$f(x) as$$ $$x$$ approaches $$a$$ from the left.

Question 3

If $$\lim\limits_{x 	o a^-} f(x) 
eq \lim\limits_{x 	o a^+} f(x)$$, what can be concluded?

Accepted Answer

The limit $$\lim\limits_{x 	o a} f(x)$$ does not exist.

Question 4

Which of the following statements is true about the existence of a limit at $$x = a$$?

Accepted Answer

The left-hand and right-hand limits must be equal.

Question 5

Suppose $$\lim\limits_{x 	o 2} f(x) = 5$$ but $$f(2) is $$undefined. What does this tell you?

Accepted Answer

The limit exists even though the function is not defined at $$x = 2.$$

Question 6

A function $$f(x) is $$defined as follows: $$f(x) = \frac{x^2$ - 4}{x - 2}$$ for $$x 
eq 2. $$What is $$\lim\limits_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 7

Which of the following is a correct interpretation of $$\lim\limits_{x 	o a^+} f(x)$$?

Accepted Answer

The value that $$f(x)$$ approaches as $$x$$ approaches $$a$$ from the right.

Question 8

If $$\lim\limits_{x 	o 3^-} f(x) = 7$$ and $$\lim\limits_{x 	o 3^+} f(x) = 7$$, what is $$\lim\limits_{x 	o 3} f(x)$$?

Accepted Answer

$$7$$

Question 9

A chemical reaction rate $$R(t) is $$modeled by a function with a removable discontinuity at $$t = 5. If$$ $$\lim\limits_{t 	o 5} R(t) = 10$$ but $$R(5) = 12$$, what does this indicate?

Accepted Answer

The limit exists and is different from the function value at $$t = 5.$$

Question 10

Which tool is NOT typically used to determine limits?

Accepted Answer

Differential equations

Question 11

Suppose $$f(x)$$ models the temperature in a chemical reaction over time. If $$\lim\limits_{x 	o 10^-} f(x) = 100$$ and $$\lim\limits_{x 	o 10^+} f(x) = 120$$, what can you conclude?

Accepted Answer

The limit $$\lim\limits_{x 	o 10} f(x)$$ does not exist.

Question 12

If $$\lim\limits_{x 	o a} f(x)$$ exists, which of the following must be true?

Accepted Answer

The left-hand and right-hand limits are equal.

Question 13

A medical dosage function $$D(t) is $$not defined at $$t = 8$$, but $$\lim\limits_{t 	o 8} D(t) = 50. $$What does this mean in practical terms?

Accepted Answer

The dosage approaches $$50 as$$ $$t$$ approaches $$8$$, even though it is not defined at $$t = 8.$$

Question 14

Which of the following is a necessary condition for $$\lim\limits_{x 	o a} f(x) to $$exist?

Accepted Answer

Both $$\lim\limits_{x 	o a^-} f(x)$$ and $$\lim\limits_{x 	o a^+} f(x)$$ exist and are equal.

Question 15

If $$\lim\limits_{x 	o 0} f(x)$$ exists but $$f(0)$$ does not, which of the following is true?

Accepted Answer

The limit exists even though the function is not defined at $$x = 0.$$

Limits: A Numerical and Graphical Approach

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