Q: For the function $h(x) = \frac{|x|}{x}$, which of the following statements is true?

$\lim_{x \to 0^-} h(x) = -1$ and $\lim_{x \to 0^+} h(x) = 1$

Q: Which property allows you to compute $\lim_{x \to c} [k \cdot f(x)]$ for a constant $k$?

$\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$

Q: Which of the following is a correct algebraic property of limits?

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

Question 1

Which of the following best describes the concept of a limit in calculus?

Accepted Answer

The value that a function f(x) approaches as x approaches a specific point c, regardless of whether f(c) is defined.

Question 2

Given $f(x) = \frac{x^2 - 4}{x - 2}$, what is $\lim_{x 	o 2} f(x)$?

Accepted Answer

$4$

Question 3

For the function $h(x) = \frac{|x|}{x}$, which of the following statements is true?

Accepted Answer

$\lim_{x 	o 0^-} h(x) = -1$ and $\lim_{x 	o 0^+} h(x) = 1$

Question 4

If $g(x) = \begin{cases} 1 + x & x 
eq 3 \ 3 & x = 3 \end{cases}$, what is $\lim_{x 	o 3} g(x)$?

Accepted Answer

$4$

Question 5

Which statement about limits is correct?

Accepted Answer

A limit can exist even if the function is not defined at that point.

Question 6

Suppose $f(x) = x^2 + 3x + 1$. What is $\lim_{x 	o 1} f(x)$?

Accepted Answer

$5$

Question 7

Which of the following is true about the limit $\lim_{x 	o c} f(x)$ for a polynomial $f(x)$?

Accepted Answer

It is always equal to $f(c)$.

Question 8

Evaluate $\lim_{x 	o 1} \sqrt{x^2 + 3x + 1}$.

Accepted Answer

$\sqrt{5}$

Question 9

If $\lim_{x 	o c} f(x) = L$ and $\lim_{x 	o c} g(x) = M$, what is $\lim_{x 	o c} [f(x) + g(x)]$?

Accepted Answer

$L + M$

Question 10

A rational function $r(x) = \frac{f(x)}{g(x)}$ has $g(c) = 0$. What can you conclude about $\lim_{x 	o c} r(x)$?

Accepted Answer

The limit does not exist unless simplification removes the zero denominator.

Question 11

Which property allows you to compute $\lim_{x 	o c} [k \cdot f(x)]$ for a constant $k$?

Accepted Answer

$\lim_{x 	o c} [k \cdot f(x)] = k \cdot \lim_{x 	o c} f(x)$

Question 12

If $\lim_{x 	o c^-} f(x) = L$ and $\lim_{x 	o c^+} f(x) = M$ with $L 
eq M$, what can you say about $\lim_{x 	o c} f(x)$?

Accepted Answer

The limit does not exist.

Question 13

A company models its profit as $P(x) = 2x^2 + 5x + 3$. What is $\lim_{x 	o 4} P(x)$?

Accepted Answer

$51$

Question 14

Which of the following is a correct algebraic property of limits?

Accepted Answer

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

Question 15

Suppose $f(x) = x^3 - 2x + 7$. What is $\lim_{x 	o 2} f(x)$?

Accepted Answer

$15$

Question 16

If $f(x) = \frac{x^2 - 9}{x - 3}$, what is $\lim_{x 	o 3} f(x)$?

Accepted Answer

$6$

Question 17

Which of the following is true for the function $f(x) = x$?

Accepted Answer

$\lim_{x 	o c} f(x) = c$

Question 18

A rational function $R(x) = \frac{x^2 - 1}{x - 1}$ is used to model a chemical reaction rate, but the denominator is zero at $x = 1$. What is $\lim_{x 	o 1} R(x)$?

Accepted Answer

$2$

Question 19

If $f(x) = 5$ for all $x$, what is $\lim_{x 	o 10} f(x)$?

Accepted Answer

$5$

Limits: Foundations and Properties in Business Calculus

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