Q: Suppose $$f(x) is $$defined for all $$x$$ near $$a$$ except possibly at $$x = a. If$$ $$\displaystyle \lim_{x \to a^-} f(x) = 3$$ and $$\displaystyle \lim_{x \to a^+} f(x) = 5$$, what can you conclude about $$\displaystyle \lim_{x \to a} f(x)$$?

$$\displaystyle \lim_{x \to a} f(x)$$ does not exist.

Q: Which of the following is the correct notation for the right-sided limit of $$f(x) as$$ $$x$$ approaches $$a$$?

$$\displaystyle \lim_{x \to a^+} f(x)$$

Q: A patient’s body temperature $$T(t) is $$measured every hour after taking medication. The values approach $$37^C as$$ $$t$$ approaches $$6$$ hours. What does $$\displaystyle \lim_{t \to 6} T(t) = 37$$ mean in this context?

The temperature gets arbitrarily close to $$37^0C as$$ $$t$$ approaches $$6$$ hours, regardless of the value at $$t = 6.$$

Q: If $$\displaystyle \lim_{x \to 3^-} f(x) = 7$$ and $$\displaystyle \lim_{x \to 3^+} f(x) = 7$$, but $$f(3) is $$undefined, what can you say about $$\displaystyle \lim_{x \to 3} f(x)$$?

$$\displaystyle \lim_{x \to 3} f(x) = 7$$

Question 1

Suppose $$f(x) is $$defined for all $$x$$ near $$a$$, except possibly at $$x = a. $$Which of the following statements best describes the meaning of $$\displaystyle \lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ gets arbitrarily close to $$L$$, regardless of the value of $$f(a).$$

Question 2

Given the following table of values for a function $$f(x)$$:

$$x$$ : $$0.9$$, $$0.99$$, $$0.999$$, $$1.1$$, $$1.01$$, $$1.001$$

$$f(x)$$ : $$0.499$$, $$0.4999$$, $$0.49999$$, $$0.501$$, $$0.5001$$, $$0.50001$$

What is the most reasonable conjecture for $$\displaystyle \lim_{x 	o 1} f(x)$$?

Accepted Answer

$$0.5$$

Question 3

Which of the following is TRUE about the relationship between one-sided and two-sided limits?

Accepted Answer

The two-sided limit exists if and only if both one-sided limits exist and are equal.

Question 4

Let $$f(x) be $$defined as follows:

$$\displaystyle f(x) = \begin{cases} x^2$ & x 
eq 2 \ 5 & x = 2 \end{cases}$$

What is $$\displaystyle \lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 5

Suppose $$f(x) is $$defined for all $$x$$ near $$a$$ except possibly at $$x = a. If$$ $$\displaystyle \lim_{x 	o a^-} f(x) = 3$$ and $$\displaystyle \lim_{x 	o a^+} f(x) = 5$$, what can you conclude about $$\displaystyle \lim_{x 	o a} f(x)$$?

Accepted Answer

$$\displaystyle \lim_{x 	o a} f(x)$$ does not exist.

Question 6

A chemical reaction's rate $$R(t) is $$modeled by a function that is undefined at $$t = 10$$ seconds, but as $$t$$ approaches $$10$$ from both sides, $$R(t)$$ approaches $$7. $$What is $$\displaystyle \lim_{t 	o 10} R(t)$$?

Accepted Answer

$$7$$

Question 7

Which of the following is the correct notation for the right-sided limit of $$f(x) as$$ $$x$$ approaches $$a$$?

Accepted Answer

$$\displaystyle \lim_{x 	o a^+} f(x)$$

Question 8

A patient’s body temperature $$T(t) is $$measured every hour after taking medication. The values approach $$37^ C as$$ $$t$$ approaches $$6$$ hours. What does $$\displaystyle \lim_{t 	o 6} T(t) = 37$$ mean in this context?

Accepted Answer

The temperature gets arbitrarily close to $$37^ 0C as$$ $$t$$ approaches $$6$$ hours, regardless of the value at $$t = 6.$$

Question 9

Let $$f(x) be $$defined for all $$x$$ near $$a$$ except possibly at $$x = a. $$Which of the following is NOT required for $$\displaystyle \lim_{x 	o a} f(x) = L to $$exist?

Accepted Answer

$$f(a) is $$defined and equal to $$L.$$

Question 10

Suppose $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 1}{x - 1}$$ for $$x 
eq 1. $$What is $$\displaystyle \lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Question 11

If $$\displaystyle \lim_{x 	o 3^-} f(x) = 7$$ and $$\displaystyle \lim_{x 	o 3^+} f(x) = 7$$, but $$f(3) is $$undefined, what can you say about $$\displaystyle \lim_{x 	o 3} f(x)$$?

Accepted Answer

$$\displaystyle \lim_{x 	o 3} f(x) = 7$$

Question 12

A function $$f(x) is $$defined for all $$x$$ near $$2$$ except at $$x = 2. If$$ $$f(x)$$ approaches different values from the left and right as $$x$$ approaches $$2$$, which of the following is true?

Accepted Answer

The two-sided limit at $$x = 2$$ does not exist.

Question 13

Suppose $$f(x) is $$defined for all $$x$$ near $$a$$ except possibly at $$x = a. If$$ $$\displaystyle \lim_{x 	o a} f(x) = L$$, which of the following must be true?

Accepted Answer

$$f(x)$$ gets arbitrarily close to $$L as$$ $$x$$ approaches $$a.$$

Question 14

A function $$f(x) is $$defined for all $$x$$ near $$0$$ except at $$x = 0. If$$ $$f(x) = \frac{1}{x}$$, what can you say about $$\displaystyle \lim_{x 	o 0} f(x)$$?

Accepted Answer

The limit does not exist.

Question 15

A function $$f(x) is $$defined for all $$x$$ near $$a$$ except possibly at $$x = a. If$$ $$\displaystyle \lim_{x 	o a} f(x)$$ does not exist, which of the following could be a possible reason?

Accepted Answer

The left and right limits as $$x$$ approaches $$a$$ are not equal.

Limits of Functions: Definitions and Numerical Approaches

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