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

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

Q: Which of the following is a correct application of the limit $$\lim_{x \to 0} \dfrac{\sin x}{x} = 1$$?

$$\lim_{x \to 0} \dfrac{\sin(3x)}{3x} = 1$$

Q: Suppose $$\lim_{x \to 2^-} f(x) = 3$$ and $$\lim_{x \to 2^+} f(x) = 5. $$Which of the following is true?

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

Question 1

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

Accepted Answer

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

Question 2

Evaluate $$\lim_{x 	o 0} \dfrac{\sin(7x)}{7x}.$$

Accepted Answer

1

Question 3

Suppose $$f(x) = \begin{cases} x-1 & x  -1 \end{cases}. $$What is $$\lim_{x 	o -1^-} f(x)$$?

Accepted Answer

$$-2$$

Question 4

Suppose $$f(x) = \begin{cases} x-1 & x  -1 \end{cases}. $$What is $$\lim_{x 	o -1^+} f(x)$$?

Accepted Answer

Question 5

Suppose $$f(x) = \begin{cases} x-1 & x  -1 \end{cases}. $$What is $$\lim_{x 	o -1} f(x)$$?

Accepted Answer

Does not exist

Question 6

Evaluate $$\lim_{x 	o 1^+} \dfrac{\sqrt{2x(x-1)}}{|x-1|}.$$

Accepted Answer

$$\sqrt{2}$$

Question 7

Evaluate $$\lim_{x 	o 1^-} \dfrac{\sqrt{2x(x-1)}}{|x-1|}.$$

Accepted Answer

Does not exist

Question 8

Evaluate $$\lim_{h 	o 0^+} \dfrac{\sqrt{h^2$ + 4h + 5} - \sqrt{5}}{h}.$$

Accepted Answer

$$\dfrac{4}{2\sqrt{5}} = \dfrac{2}{\sqrt{5}}$$

Question 9

Evaluate $$\lim_{	heta 	o 0} \dfrac{\sin(\sqrt{2}	heta)}{\sqrt{2}	heta}.$$

Accepted Answer

1

Question 10

Evaluate $$\lim_{x 	o 0} \dfrac{	an(2x)}{x}.$$

Accepted Answer

2

Question 11

Evaluate $$\lim_{x 	o 0} \dfrac{x^2$ - x + \sin x}{2x}.$$

Accepted Answer

$$-\dfrac{1}{2}$$

Question 12

Evaluate $$\lim_{x 	o 0} \dfrac{\sin(5x)}{\sin(4x)}.$$

Accepted Answer

$$\dfrac{5}{4}$$

Question 13

A medical device measures the angle $$	heta (in $$radians) of a joint as it moves. If $$	heta is $$very small, which of the following best approximates $$\sin 	heta$$?

Accepted Answer

$$\sin 	heta \approx 	heta$$

Question 14

Which of the following is a correct application of the limit $$\lim_{x 	o 0} \dfrac{\sin x}{x} = 1$$?

Accepted Answer

$$\lim_{x 	o 0} \dfrac{\sin(3x)}{3x} = 1$$

Question 15

A chemical reaction's rate is modeled by $$f(x) = \dfrac{\sin(2x)}{x}$$ for small $$x. $$What is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

2

Question 16

Which of the following is the correct definition of the right-hand limit $$\lim_{x 	o c^+} f(x)$$?

Accepted Answer

The value $$f(x)$$ approaches as $$x$$ approaches $$c$$ from values greater than $$c$$

Question 17

Suppose $$\lim_{x 	o 2^-} f(x) = 3$$ and $$\lim_{x 	o 2^+} f(x) = 5. $$Which of the following is true?

Accepted Answer

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

Question 18

A car's position is modeled by $$s(t) = \sin(2t)$$ for small $$t. $$What is $$\lim_{t 	o 0} \dfrac{s(t)}{t}$$?

Accepted Answer

2

Question 19

Which of the following best describes the meaning of $$\lim_{x 	o c} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$c$$, $$f(x)$$ approaches $$L$$ regardless of $$f(c)$$

Calculus I: Limits and Limit Laws – Step-by-Step Study Guidance

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