Q: If $$\lim\limits_{x \to 2^-} f(x) = 5$$ and $$\lim\limits_{x \to 2^+} f(x) = 7$$, what can you conclude about $$\lim\limits_{x \to 2} f(x)$$?

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

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

$$\lim\limits_{x \to a} f(x) = L$$

Question 1

Which of the following best describes the meaning of $$\lim\limits_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ becomes arbitrarily close to $$L$$, but $$x$$ does not have to equal $$a.$$

Question 2

Given the function $$f(x) = \dfrac{x^2$ - 9}{x - 3}$$, what is $$\lim\limits_{x 	o 3} f(x)$$?

Accepted Answer

$$6$$

Question 3

For the function $$f(x) = \dfrac{1}{x^2$} + 3$$, what is $$\lim\limits_{x 	o \infty} f(x)$$?

Accepted Answer

$$3$$

Question 4

Suppose $$H(x) = \begin{cases} 2x+2 & 	ext{if } x\u003c1 \ 2x-4 & 	ext{if } x \geq 1 \end{cases}. $$What is $$\lim\limits_{x 	o 1^-} H(x)$$?

Accepted Answer

$$4$$

Question 5

Suppose $$H(x) = \begin{cases} 2x+2 & 	ext{if } x\u003c1 \ 2x-4 & 	ext{if } x \geq 1 \end{cases}. $$What is $$\lim\limits_{x 	o 1^+} H(x)$$?

Accepted Answer

$$-2$$

Question 6

Given $$H(x) as $$above, does $$\lim\limits_{x 	o 1} H(x)$$ exist?

Accepted Answer

No, because the left and right limits are not equal.

Question 7

A chemical reaction's temperature $$T (in$$ $$^0C) is $$modeled by $$T(x) = \dfrac{1}{x^2$} + 3$$, where $$x is $$time in minutes. What is the long-term temperature as $$x 	o \infty$$?

Accepted Answer

$$3$$

Question 8

If $$\lim\limits_{x 	o 2^-} f(x) = 5$$ and $$\lim\limits_{x 	o 2^+} f(x) = 7$$, 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 9

A function $$f(x) is $$defined everywhere except at $$x=4. If$$ $$\lim\limits_{x 	o 4} f(x) = 10$$, which of the following must be true?

Accepted Answer

As $$x$$ approaches $$4$$, $$f(x)$$ gets arbitrarily close to $$10.$$

Question 10

Consider the sequence $$5.51, 5.501, 5.5001, 5.50001, \ldots. $$Which notation best describes the limit of this sequence?

Accepted Answer

$$x 	o 5.5^+$$

Question 11

Suppose $$f(x) is $$defined as $$f(x) = \dfrac{1}{x^2$} + 3. $$What is $$\lim\limits_{x 	o 0^+} f(x)$$?

Accepted Answer

$$\infty$$

Question 12

A business models its profit $$P(x) as$$ $$P(x) = \dfrac{x^2$ - 9}{x - 3}. $$What is the profit as $$x$$ approaches $$3$$?

Accepted Answer

$$6$$

Question 13

If $$\lim\limits_{x 	o a^-} f(x) = L$$ and $$\lim\limits_{x 	o a^+} f(x) = L$$, what can you conclude?

Accepted Answer

$$\lim\limits_{x 	o a} f(x) = L$$

Question 14

A function $$h(x)$$ has $$\lim\limits_{x 	o 2^-} h(x) = 5$$ and $$\lim\limits_{x 	o 2^+} h(x) = 5. $$What is $$\lim\limits_{x 	o 2} h(x)$$?

Accepted Answer

$$5$$

Question 15

Suppose $$f(x) is $$defined everywhere, but $$f(a) is $$not defined. If $$\lim\limits_{x 	o a} f(x)$$ exists, what does this mean?

Accepted Answer

The limit can exist even if $$f(a) is $$undefined.

Question 16

A medical dosage function $$D(x) is $$defined as $$D(x) = \dfrac{1}{x^2$} + 3. $$What happens to $$D(x) as$$ $$x$$ approaches $$0$$ from the right?

Accepted Answer

$$D(x)$$ approaches $$\infty$$

Question 17

If a function $$f(x)$$ has $$\lim\limits_{x 	o 2^-} f(x) = 3$$ and $$\lim\limits_{x 	o 2^+} f(x) = 3$$, but $$f(2) = 5$$, what is $$\lim\limits_{x 	o 2} f(x)$$?

Accepted Answer

$$3$$

Question 18

Which of the following is true about the existence of a limit at $$x=a$$?

Accepted Answer

The left-hand and right-hand limits at $$x=a$$ must be equal.

Question 19

A function $$f(x) is $$given by $$f(x) = \dfrac{1}{x^2$} + 3. $$What is $$\lim\limits_{x 	o -\infty} f(x)$$?

Accepted Answer

$$3$$

Limits: A Numerical and Graphical Approach (Business Calculus Study Notes)

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