Q: Suppose the sequence $$2.24, 2.249, 2.2499, 2.24999, \ldots is $$given. What is the limit of this sequence as $$x$$ increases, and how should it be written using limit notation?

$$\lim\limits_{x \to 2.25^-} = 2.25$$

Q: For the function $$f(x) = \dfrac{x^2$ - 9}{x - 3}$$, what is $$\lim\limits_{x \to 3} f(x)$$?

$$\lim\limits_{x \to 3} f(x) = 6$$

Q: Consider the piecewise function $$H(x) = \begin{cases} 2x+2 & \text{if } x\u003c1 \\ 2x-4 & \text{if } x \geq 1 \end{cases}. $$What is $$\lim\limits_{x \to 1} H(x)$$?

$$\lim\limits_{x \to 1} H(x)$$ does not exist

Q: For the same function $$H(x) as $$above, what is $$\lim\limits_{x \to -3} H(x)$$?

$$\lim\limits_{x \to -3} H(x) = -4$$

Q: Given $$f(x) = \dfrac{1}{x^2$} + 3$$, what is $$\lim\limits_{x \to 3} f(x)$$?

$$\lim\limits_{x \to 3} f(x) = 4$$

Q: For the function $$f(x) = \dfrac{1}{x^2$} + 3$$, what is $$\lim\limits_{x \to 0^+} f(x)$$?

$$\lim\limits_{x \to 0^+} f(x) = \infty$$

Q: A function $$f(x) is $$defined such that $$\lim\limits_{x \to 2^-} f(x) = 5$$ and $$\lim\limits_{x \to 2^+} f(x) = 7. $$What can you conclude about $$\lim\limits_{x \to 2} f(x)$$?

$$\lim\limits_{x \to 2} f(x)$$ does not exist

Q: Suppose $$h(x) = \dfrac{1}{x-1} + 6. $$What is $$\lim\limits_{x \to 1} h(x)$$?

$$\lim\limits_{x \to 1} h(x)$$ does not exist

Q: For the function $$h(x) = \dfrac{1}{x-1} + 6$$, what is $$\lim\limits_{x \to 2} h(x)$$?

$$\lim\limits_{x \to 2} h(x) = 7$$

Question 1

Suppose the sequence $$2.24, 2.249, 2.2499, 2.24999, \ldots is $$given. What is the limit of this sequence as $$x$$ increases, and how should it be written using limit notation?

Accepted Answer

$$\lim\limits_{x 	o 2.25^-} = 2.25$$

Question 2

Given the function $$f(x) = \dfrac{x^2$ - 9}{x - 3}$$, what is $$f(3)$$?

Accepted Answer

$$f(3)$$ does not exist$

Question 3

For the function $$f(x) = \dfrac{x^2$ - 9}{x - 3}$$, what is $$\lim\limits_{x 	o 3} f(x)$$?

Accepted Answer

$$\lim\limits_{x 	o 3} f(x) = 6$$

Question 4

Which of the following statements is TRUE regarding the existence of $$\lim\limits_{x 	o a} f(x)$$?

Accepted Answer

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

Question 5

Consider the piecewise function $$H(x) = \begin{cases} 2x+2 & 	ext{if } x\u003c1 \ 2x-4 & 	ext{if } x \geq 1 \end{cases}. $$What is $$\lim\limits_{x 	o 1} H(x)$$?

Accepted Answer

$$\lim\limits_{x 	o 1} H(x)$$ does not exist

Question 6

For the same function $$H(x) as $$above, what is $$\lim\limits_{x 	o -3} H(x)$$?

Accepted Answer

$$\lim\limits_{x 	o -3} H(x) = -4$$

Question 7

Given $$f(x) = \dfrac{1}{x^2$} + 3$$, what is $$\lim\limits_{x 	o 3} f(x)$$?

Accepted Answer

$$\lim\limits_{x 	o 3} f(x) = 4$$

Question 8

For the function $$f(x) = \dfrac{1}{x^2$} + 3$$, what is $$\lim\limits_{x 	o 0^+} f(x)$$?

Accepted Answer

$$\lim\limits_{x 	o 0^+} f(x) = \infty$$

Question 9

A function $$f(x) is $$defined such that $$\lim\limits_{x 	o 2^-} f(x) = 5$$ and $$\lim\limits_{x 	o 2^+} f(x) = 7. $$What can you conclude about $$\lim\limits_{x 	o 2} f(x)$$?

Accepted Answer

$$\lim\limits_{x 	o 2} f(x)$$ does not exist

Question 10

Suppose $$h(x) = \dfrac{1}{x-1} + 6. $$What is $$\lim\limits_{x 	o 1} h(x)$$?

Accepted Answer

$$\lim\limits_{x 	o 1} h(x)$$ does not exist

Question 11

For the function $$h(x) = \dfrac{1}{x-1} + 6$$, what is $$\lim\limits_{x 	o 2} h(x)$$?

Accepted Answer

$$\lim\limits_{x 	o 2} h(x) = 7$$

Question 12

Evaluate $$\lim\limits_{x 	o \infty} \left( \dfrac{1}{x-1} + 6 ight). $$What is the result?

Accepted Answer

$$\lim\limits_{x 	o \infty} h(x) = 6$$

Question 13

A company’s cost function is $$C(x) = \dfrac{1}{x^2$} + 3$$, where $$x is $$the number of units produced. What happens to the cost per unit as production increases without bound?

Accepted Answer

The cost per unit approaches $$3$$

Question 14

If $$\lim\limits_{x 	o a^-} f(x) = L$$ and $$\lim\limits_{x 	o a^+} f(x) = L$$, but $$f(a) is $$undefined, what can you say about $$\lim\limits_{x 	o a} f(x)$$?

Accepted Answer

$$\lim\limits_{x 	o a} f(x) = L$$

Question 15

A function $$f(x) is $$defined such that $$f(2)$$ does not exist, but $$\lim\limits_{x 	o 2} f(x) = 5. $$Which of the following is true?

Accepted Answer

The limit exists even though the function is undefined at $$x=2$$

Question 16

Which of the following best describes the 'wall method' for finding limits graphically?

Accepted Answer

Draw a vertical line at $$x = a$$ and observe the $$y-$$values approached from both sides.

Question 17

Suppose $$f(x) is $$defined for all $$x$$ except $$x = 4$$, and $$\lim\limits_{x 	o 4^-} f(x) = 2$$, $$\lim\limits_{x 	o 4^+} f(x) = 2. $$What is $$\lim\limits_{x 	o 4} f(x)$$?

Accepted Answer

$$\lim\limits_{x 	o 4} f(x) = 2$$

Limits: A Numerical and Graphical Approach (Business Calculus Study Notes)

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