Q: Which of the following statements is TRUE about the existence of $$\displaystyle \lim_{x \to c} f(x)$$?

$$\displaystyle \lim_{x \to c} f(x)$$ exists if and only if $$\displaystyle \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$$

Q: Given $$f(x) = \dfrac{x^2$ + x - 6}{|x-2|}$$, does $$\displaystyle \lim_{x \to 2} f(x)$$ exist?

$$\text{No, because the left and right limits are not equal}$$

Q: A function $$f(x) is $$defined everywhere except at $$x=5. $$The left-hand limit as $$x \to 5 is$$ $$2$$, and the right-hand limit as $$x \to 5 is$$ $$3. $$What can you conclude about $$\displaystyle \lim_{x \to 5} f(x)$$?

$$\displaystyle \lim_{x \to 5} f(x)$$ does not exist

Q: Suppose a function $$f(x) is $$defined such that $$\displaystyle \lim_{x \to 2^-} f(x) = 5$$ and $$\displaystyle \lim_{x \to 2^+} f(x) = 5. $$Which of the following must be true?

$$\displaystyle \lim_{x \to 2} f(x) = 5$$

Q: Suppose $$f(x) is $$defined for all $$x$$ except $$x=0. If$$ $$\displaystyle \lim_{x \to 0^-} f(x) = 3$$ and $$\displaystyle \lim_{x \to 0^+} f(x) = 3$$, but $$f(0) is $$undefined, what can you say about $$\displaystyle \lim_{x \to 0} f(x)$$?

$$\displaystyle \lim_{x \to 0} f(x) = 3$$

Question 1

Let $$f(x) = \dfrac{x}{|x|}. $$What is $$\displaystyle \lim_{x 	o 0^-} f(x)$$?

Accepted Answer

$$-1$$

Question 2

Given $$f(x) = \dfrac{x}{|x|}$$, what is $$\displaystyle \lim_{x 	o 0} f(x)$$?

Accepted Answer

$$	ext{Does not exist}$$

Question 3

Which of the following statements is TRUE about the existence of $$\displaystyle \lim_{x 	o c} f(x)$$?

Accepted Answer

$$\displaystyle \lim_{x 	o c} f(x)$$ exists if and only if $$\displaystyle \lim_{x 	o c^-} f(x) = \lim_{x 	o c^+} f(x)$$

Question 4

Suppose $$g(x) is $$defined as $$g(x) = \begin{cases} x-4 & x \u003e 4 \ 8-2x & x \u003c 4 \end{cases}. $$What is $$\displaystyle \lim_{x 	o 4^-} g(x)$$?

Accepted Answer

$$8-2 	imes 4 = 0$$

Question 5

For the same function $$g(x) = \begin{cases} x-4 & x \u003e 4 \ 8-2x & x \u003c 4 \end{cases}$$, what is $$\displaystyle \lim_{x 	o 4^+} g(x)$$?

Accepted Answer

$$0$$

Question 6

Given $$f(x) = \dfrac{x^2$ + x - 6}{|x-2|}$$, does $$\displaystyle \lim_{x 	o 2} f(x)$$ exist?

Accepted Answer

$$	ext{No, because the left and right limits are not equal}$$

Question 7

Suppose $$h(x) is $$defined as follows: $$h(x) = \begin{cases} 2x+1 & x  3 \end{cases}. $$What is $$\displaystyle \lim_{x 	o 3^-} h(x)$$?

Accepted Answer

$$2 	imes 3 + 1 = 7$$

Question 8

For the function $$h(x) in $$the previous question, what is $$\displaystyle \lim_{x 	o 3^+} h(x)$$?

Accepted Answer

$$3^2 = 9$

Question 9

A function $$f(x) is $$defined everywhere except at $$x=5. $$The left-hand limit as $$x 	o 5 is$$ $$2$$, and the right-hand limit as $$x 	o 5 is$$ $$3. $$What can you conclude about $$\displaystyle \lim_{x 	o 5} f(x)$$?

Accepted Answer

$$\displaystyle \lim_{x 	o 5} f(x)$$ does not exist

Question 10

Suppose a function $$f(x) is $$defined such that $$\displaystyle \lim_{x 	o 2^-} f(x) = 5$$ and $$\displaystyle \lim_{x 	o 2^+} f(x) = 5. $$Which of the following must be true?

Accepted Answer

$$\displaystyle \lim_{x 	o 2} f(x) = 5$$

Question 11

A chemical reaction rate $$R(t) is $$modeled by a piecewise function: $$R(t) = \begin{cases} 10-t & t  2 \end{cases}. $$What is $$\displaystyle \lim_{t 	o 2^-} R(t)$$?

Accepted Answer

$$10-2 = 8$$

Question 12

For the same reaction rate $$R(t)$$, what is $$\displaystyle \lim_{t 	o 2^+} R(t)$$?

Accepted Answer

$$2^2 = 4$

Question 13

A patient’s heart rate $$H(t) is $$modeled by $$H(t) = \begin{cases} 70 & t  1 \end{cases}. $$What is $$\displaystyle \lim_{t 	o 1} H(t)$$?

Accepted Answer

$$	ext{Does not exist}$$

Question 14

Suppose $$f(x) is $$defined for all $$x$$ except $$x=0. If$$ $$\displaystyle \lim_{x 	o 0^-} f(x) = 3$$ and $$\displaystyle \lim_{x 	o 0^+} f(x) = 3$$, but $$f(0) is $$undefined, what can you say about $$\displaystyle \lim_{x 	o 0} f(x)$$?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} f(x) = 3$$

Question 15

A function $$g(x) is $$defined as $$g(x) = \begin{cases} 2x & x  1 \end{cases}. $$What is $$\displaystyle \lim_{x 	o 1} g(x)$$?

Accepted Answer

$$1 + 1 = 2$$

Question 16

Suppose $$f(x) is $$defined as $$f(x) = \dfrac{x^2$ + x - 6}{x-2}$$ for $$x 
eq 2. $$What is $$\displaystyle \lim_{x 	o 2} f(x)$$?

Accepted Answer

$$2 + 3 = 5 ($$since $$x^2 + x - 6 = (x-2)(x+3$$)$$, so f(x) = x+3$ for x$$ 
eq 2$$; thus, $$\lim_{x 	o 2} f(x) = 5$$)

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