Q: If $$\lim\limits_{x \to 2^-} f(x) = 1$$ and $$\lim\limits_{x \to 2^+} f(x) = 3$$, what can you conclude about $$\lim\limits_{x \to 2} f(x)$$?

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

Q: A function $$f(x) is $$continuous at $$x = a if$$:

$$\lim\limits_{x \to a} f(x)$$ exists and equals $$f(a)$$

Q: Given the piecewise function $$f(x) = \begin{cases} x+1, & x \u003c 2 \\ 2x-1, & x \geq 2 \end{cases}$$, is $$f(x)$$ continuous at $$x = 2$$?

Yes, because $$\lim\limits_{x \to 2^-} f(x) = \lim\limits_{x \to 2^+} f(x) = f(2)$$

Q: Which of the following limit laws is correct?

$$\lim\limits_{x \to a} [f(x) + g(x)] = \lim\limits_{x \to a} f(x) + \lim\limits_{x \to a} g(x)$$

Question 1

Which of the following statements best describes the limit of a function $$f(x) as$$ $$x$$ approaches $$a$$?

Accepted Answer

It is the value that $$f(x)$$ gets arbitrarily close to as $$x$$ gets close to $$a$$, but not necessarily equal to $$a.$$

Question 2

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

Accepted Answer

$$3$$

Question 3

If $$\lim\limits_{x 	o 2^-} f(x) = 1$$ and $$\lim\limits_{x 	o 2^+} f(x) = 3$$, 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 4

A function $$f(x) is $$continuous at $$x = a if$$:

Accepted Answer

$$\lim\limits_{x 	o a} f(x)$$ exists and equals $$f(a)$$

Question 5

Which of the following is an example of a jump discontinuity?

Accepted Answer

The graph of $$f(x)$$ has a sudden jump from one value to another at $$x = c.$$

Question 6

Evaluate $$\lim\limits_{x 	o 0^-} f(x)$$ and $$\lim\limits_{x 	o 0^+} f(x)$$ for a function where $$f(x)$$ approaches $$2$$ from the left and $$4$$ from the right as $$x$$ approaches $$0. $$What is $$\lim\limits_{x 	o 0} f(x)$$?

Accepted Answer

Does not exist

Question 7

Suppose $$f(x) is $$defined as $$f(x) = \sqrt{x^2$ - 4x}$$ for all $$x \geq 4. $$What is $$\lim\limits_{x 	o 4} f(x)$$?

Accepted Answer

$$0$$

Question 8

Which of the following is true about the left-hand and right-hand limits?

Accepted Answer

If the left-hand and right-hand limits at $$x = a$$ are equal, then $$\lim\limits_{x 	o a} f(x)$$ exists.

Question 9

Given the piecewise function $$f(x) = \begin{cases} x+1, & x \u003c 2 \ 2x-1, & x \geq 2 \end{cases}$$, is $$f(x)$$ continuous at $$x = 2$$?

Accepted Answer

Yes, because $$\lim\limits_{x 	o 2^-} f(x) = \lim\limits_{x 	o 2^+} f(x) = f(2)$$

Question 10

Which of the following limit laws is correct?

Accepted Answer

$$\lim\limits_{x 	o a} [f(x) + g(x)] = \lim\limits_{x 	o a} f(x) + \lim\limits_{x 	o a} g(x)$$

Question 11

A business analyst models the cost function $$C(x)$$ for producing $$x$$ units. If $$\lim\limits_{x 	o 100} C(x) = 5000$$, what does this mean in practical terms?

Accepted Answer

As the number of units produced gets very close to 100, the cost approaches $5000.

Question 12

If $$\lim\limits_{x 	o 3} f(x) = 2$$ and $$f(3) = 5$$, what can you say about the continuity of $$f(x) at$$ $$x = 3$$?

Accepted Answer

$$f(x) is $$not continuous at $$x = 3.$$

Question 13

Which of the following is NOT a requirement for a function $$f(x) to be $$continuous at $$x = a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = a$$

Question 14

Evaluate $$\lim\limits_{x 	o 2} \sqrt{x^2$ - 4x + 4}.$$

Accepted Answer

$$0$$

Question 15

A function $$f(x)$$ has a hole at $$x = 5. $$What does this mean?

Accepted Answer

$$f(x) is $$not defined at $$x = 5$$, but $$\lim\limits_{x 	o 5} f(x)$$ exists.

Question 16

If $$\lim\limits_{x 	o a} f(x) = L$$ and $$\lim\limits_{x 	o a} g(x) = M$$, what is $$\lim\limits_{x 	o a} [f(x) \cdot g(x)]$$?

Accepted Answer

$$L \cdot M$$

Question 17

A company’s revenue function $$R(x) is $$continuous at $$x = 100. $$What does this imply about $$R(x) at$$ $$x = 100$$?

Accepted Answer

$$\lim\limits_{x 	o 100} R(x) = R(100)$$

Business Calculus: Limits and Continuity (Section 1.1 Study Notes)

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