Q: Suppose $$\lim_{x \to 3} f(x) = \text{DNE}. $$Which of the following could be a reason?

The left and right limits as $$x \to 3$$ are not equal.

Question 1

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

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ gets arbitrarily close to $$L$$, regardless of $$f(a).$$

Question 2

Suppose $$f(2) = 	ext{DNE}$$ and $$\lim_{x 	o 2} f(x) = 4. $$Which of the following is true?

Accepted Answer

$$f(x)$$ approaches 4 as $$x$$ approaches 2, even though $$f(2) is $$undefined.

Question 3

Given the graph of $$g(x)$$ with $$g(2) = 6$$ and $$\lim_{x 	o 2} g(x) = 4$$, what can you conclude?

Accepted Answer

$$g(x)$$ has a removable discontinuity at $$x=2.$$

Question 4

If $$\lim_{x 	o 1^-} g(x) = 2$$ and $$\lim_{x 	o 1^+} g(x) = 3$$, what is $$\lim_{x 	o 1} g(x)$$?

Accepted Answer

$$	ext{DNE}$$

Question 5

A function $$f(x) is $$defined as follows: $$f(x) = \frac{x^2$ - 4}{x-2}$$ for $$x 
eq 2. $$What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 6

Suppose $$\lim_{x 	o 3} f(x) = 	ext{DNE}. $$Which of the following could be a reason?

Accepted Answer

The left and right limits as $$x 	o 3$$ are not equal.

Question 7

Given the graph of $$g(x)$$ with $$g(-1) = 3$$ and $$\lim_{x 	o -1} g(x) = 2$$, which of the following statements is correct?

Accepted Answer

$$g(x)$$ has a jump discontinuity at $$x = -1.$$

Question 8

If $$\lim_{x 	o 2^-} f(x) = 4$$ and $$\lim_{x 	o 2^+} f(x) = 1$$, what is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$	ext{DNE}$$

Question 9

A patient’s medication concentration in the bloodstream is modeled by $$C(t)$$, where $$t is $$time in hours. If $$\lim_{t 	o 5} C(t) = 2 mg/L$$, what does this mean in a medical context?

Accepted Answer

As time approaches $$5$$ hours, the concentration approaches $$2 mg/L$$, regardless of the value at $$t=5.$$

Question 10

Suppose $$\lim_{x 	o 3} g(x) = 4$$ and $$g(3) = 2. $$Which of the following is true?

Accepted Answer

$$g(x)$$ has a removable discontinuity at $$x=3.$$

Question 11

If $$\lim_{x 	o 1^-} g(x) = 2$$ and $$\lim_{x 	o 1^+} g(x) = 2$$, but $$g(1) = 5$$, what type of discontinuity does $$g(x)$$ have at $$x=1$$?

Accepted Answer

Removable discontinuity

Question 12

A chemical reaction rate $$R(t) is $$undefined at $$t=10$$ seconds, but $$\lim_{t 	o 10} R(t) = 7. $$What does this imply about the reaction at $$t=10$$?

Accepted Answer

The reaction rate approaches $$7 as$$ $$t$$ approaches $$10$$ seconds, even though it is undefined at $$t=10.$$

Question 13

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

Accepted Answer

$$4$$

Question 14

If $$\lim_{x 	o 3^-} g(x) = 4$$ and $$\lim_{x 	o 3^+} g(x) = 5$$, what is $$\lim_{x 	o 3} g(x)$$?

Accepted Answer

$$	ext{DNE}$$

Question 15

A function $$h(x) is $$defined for all $$x$$ except $$x=1. If$$ $$\lim_{x 	o 1} h(x)$$ exists, which of the following must be true?

Accepted Answer

$$h(x)$$ approaches a single value as $$x$$ approaches $$1$$ from both sides.

Question 16

Suppose $$\lim_{x 	o 2} f(x) = 4$$ and $$\lim_{x 	o 2} g(x) = 5. $$What is $$\lim_{x 	o 2} [f(x) + g(x)]$$?

Accepted Answer

$$9$$

Question 17

If $$\lim_{x 	o 1} g(x) = 2$$ and $$g(1) = 5$$, which of the following is true?

Accepted Answer

$$g(x)$$ has a removable discontinuity at $$x=1.$$

Question 18

A function $$f(x) is $$defined as $$f(x) = \begin{cases} 2x+1 & x  3 \end{cases}. $$What is $$\lim_{x 	o 3} f(x)$$?

Accepted Answer

$$	ext{DNE}$$

Question 19

Suppose $$\lim_{x 	o 5^-} g(x) = 5$$ and $$\lim_{x 	o 5^+} g(x) = 5. $$What is $$\lim_{x 	o 5} g(x)$$?

Accepted Answer

$$5$$

Definition of Limits and One-Sided Limits 2.2

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