$$\lim_{x \to c} [f(x)] = \lim_{x \to c} [g(x)]$$ for any functions $$f$$ and $$g$$

$$\lim_{x \to 2} f(x) = 5$$ and $$\lim_{x \to 2} g(x) = 1$$

$$\lim_{x \to 2^-} f(x) = 5$$ and $$\lim_{x \to 2^+} f(x) = 1$$

Question 1

Which of the following best describes the difference between algebra and calculus in terms of their focus?

Accepted Answer

Algebra solves equations for particular values (static), while calculus studies how changes in one variable affect another (dynamic).

Question 2

Who were the two mathematicians credited with the independent development of calculus?

Accepted Answer

Gottfried Wilhelm Leibniz and Isaac Newton

Question 3

Which concept is foundational to both the derivative and the integral in calculus?

Accepted Answer

The concept of a limit

Question 4

Given the function $$f(x) = 2x - 1$$, what is the value of $$f(-1)$$?

Accepted Answer

$$f(-1) = -3$$

Question 5

If $$f(x) = x + 2$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 6

Which statement about the existence of a limit is correct?

Accepted Answer

A limit exists at $$x = c if $$the function is defined on both sides of $$c$$, regardless of $$f(c).$$

Question 7

Consider the absolute value function $$h(x) = |x|. $$What is $$\lim_{x 	o 0} h(x)$$?

Accepted Answer

$$0$$

Question 8

If the left-hand limit and right-hand limit of $$f(x) as$$ $$x$$ approaches $$c$$ are not equal, what can be concluded?

Accepted Answer

The two-sided limit does not exist.

Question 9

Evaluate $$\lim_{x 	o 2} (x^2$ - 3x)$$ using limit properties.

Accepted Answer

$$-2$$

Question 10

Which of the following is NOT a property of limits?

Accepted Answer

$$\lim_{x 	o c} [f(x)] = \lim_{x 	o c} [g(x)]$$ for any functions $$f$$ and $$g$$

Question 11

For the function $$f(x) = x^2$ + 1$$ and $$g(x) = x - 1$$, what is $$\lim_{x 	o 2} f(x)$$ and $$\lim_{x 	o 2} g(x)$$?

Accepted Answer

$$\lim_{x 	o 2} f(x) = 5$$ and $$\lim_{x 	o 2} g(x) = 1$$

Question 12

Suppose $$\lim_{x 	o c} f(x) = 0$$ and $$\lim_{x 	o c} g(x) = 0. $$What is the form of $$\lim_{x 	o c} \frac{f(x)}{g(x)}$$?

Accepted Answer

It is an indeterminate form ($$0/0).$$

Question 13

Evaluate $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 14

If $$\lim_{x 	o c} f(x) = L 
eq 0$$ and $$\lim_{x 	o c} g(x) = 0$$, what can be said about $$\lim_{x 	o c} \frac{f(x)}{g(x)}$$?

Accepted Answer

The limit does not exist.

Question 15

Find $$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$ for $$f(x) = 2x - 3$$ and $$a = 1.$$

Accepted Answer

$$2$$

Question 16

A pharmaceutical company models the cost of producing $$x$$ units of a drug by $$C(x) = 5x^2$ + 20x + 100. $$What is $$\lim_{x 	o 10} C(x)$$?

Accepted Answer

$$C(10) = 800$$

Question 17

A company’s revenue function is $$R(x) = 3x^2$ - 2x + 5. $$What is $$\lim_{x 	o 4} R(x)$$?

Accepted Answer

$$R(4) = 49$$

Question 18

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

Accepted Answer

$$\lim_{x 	o 2^-} f(x) = 5$$ and $$\lim_{x 	o 2^+} f(x) = 1$$

Limits and Introduction to Calculus for Business, Economics, and Life Sciences

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