$$\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$$

If $$g(x) \leq f(x) \leq h(x)$$ near $$c$$ and $$\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$$, then $$\lim_{x \to c} f(x) = L$$

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} if$$ $$\lim_{x \to c} g(x) \neq 0$$

$$\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$$

$$\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)}$$

Question 1

Based on the table provided, what is the value of $$\lim_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$1$$

Question 2

Which of the following is the correct application of the Sum Rule for limits?

Accepted Answer

$$\lim_{x 	o c} [f(x) + g(x)] = \lim_{x 	o c} f(x) + \lim_{x 	o c} g(x)$$

Question 3

Evaluate $$\lim_{x 	o 2} \frac{2x + 5}{11 - x^3$}.$$

Accepted Answer

Question 4

Which theorem allows you to evaluate $$\lim_{x 	o c} P(x)$$ for a polynomial $$P(x) by $$direct substitution?

Accepted Answer

Theorem 2 (Limits of Polynomials)

Question 5

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

Accepted Answer

$$2$$

Question 6

Which of the following is NOT a requirement for applying Theorem 3 (Limits of Rational Functions)?

Accepted Answer

$$P(x)$$ must be a constant function$

Question 7

Evaluate $$\lim_{x 	o -3} (x^2$ - 13).$$

Accepted Answer

$$-4$$

Question 8

Which of the following best describes the Sandwich (Squeeze) Theorem?

Accepted Answer

If $$g(x) \leq f(x) \leq h(x)$$ near $$c$$ and $$\lim_{x 	o c} g(x) = \lim_{x 	o c} h(x) = L$$, then $$\lim_{x 	o c} f(x) = L$$

Question 9

Find $$\lim_{x 	o 0} x^2$ \sin \left(\frac{1}{x}ight)$$ using the Sandwich Theorem.

Accepted Answer

$$0$$

Question 10

Which of the following is a correct application of the Quotient Rule for limits?

Accepted Answer

$$\lim_{x 	o c} \frac{f(x)}{g(x)} = \frac{\lim_{x 	o c} f(x)}{\lim_{x 	o c} g(x)} if$$ $$\lim_{x 	o c} g(x) 
eq 0$$

Question 11

Evaluate $$\lim_{x 	o -1} \frac{x^2$ + 8 - 3}{x + 1}.$$

Accepted Answer

$$4$$

Question 12

Which of the following statements is TRUE about limits?

Accepted Answer

The limit as $$x$$ approaches $$c$$ depends only on the values of $$f(x)$$ near $$c$$, not at $$c$$ itself

Question 13

Evaluate $$\lim_{	heta 	o 0} (2 \sin 	heta - 1).$$

Accepted Answer

$$-1$$

Question 14

Which of the following is a correct application of the Power Rule for limits?

Accepted Answer

$$\lim_{x 	o c} [f(x)]^n = [\lim_{x 	o c} f(x)]^n$$

Question 15

A medical researcher is modeling the concentration of a drug in the bloodstream as $$C(t) = \frac{10}{t^2$ + 1}. $$What is $$\lim_{t 	o 0} C(t)$$?

Accepted Answer

$$10$$

Question 16

Evaluate $$\lim_{x 	o 0} \left( \frac{1}{x-1} + \frac{1}{x+1} ight) x.$$

Accepted Answer

$$0$$

Question 17

Which of the following is a correct application of the Root Rule for limits?

Accepted Answer

$$\lim_{x 	o c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x 	o c} f(x)}$$

2.2

Guida di studio - Domande di pratica