Q: Suppose $$f(x) is $$defined as $$f(x) = \frac{2x^2$ - 6x + 4}{|x-1|}. If$$ $$\lim_{x \to 1^+} f(x) = -2$$ and $$\lim_{x \to 1^-} f(x) = 2$$, what can you conclude about $$\lim_{x \to 1} f(x)$$?

$$\lim_{x \to 1} f(x)$$ does not exist

Question 1

Given the function $$f(x) = x^2$ - x + 1$$, use the table of values approaching $$x = 2$$ from both sides to estimate $$\lim_{x 	o 2} f(x). $$What is the value of this limit?

Accepted Answer

$$3$$

Question 2

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

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ becomes arbitrarily close to $$L$$ from both sides of $$a.$$

Question 3

Consider the function $$f(x) = \frac{\sqrt{x} - 1}{x - 1}. $$Based on the table of values as $$x$$ approaches $$1$$, what is $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$2$$

Question 4

Which of the following is true about the value of $$f(a)$$ and $$\lim_{x 	o a} f(x)$$?

Accepted Answer

The limit describes the behavior near $$a$$, not necessarily at $$a.$$

Question 5

Suppose $$f(x) is $$defined as $$f(x) = \frac{2x^2$ - 6x + 4}{|x-1|}. If$$ $$\lim_{x 	o 1^+} f(x) = -2$$ and $$\lim_{x 	o 1^-} f(x) = 2$$, what can you conclude about $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$\lim_{x 	o 1} f(x)$$ does not exist

Question 6

Which of the following is the correct definition of the right-sided limit $$\lim_{x 	o a^+} f(x)$$?

Accepted Answer

The value $$f(x)$$ approaches as $$x$$ approaches $$a$$ from the right.

Question 7

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

Accepted Answer

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

Question 8

Consider the function $$f(x) = \frac{\sqrt{x} - 1}{x - 1}. $$Why is there a 'hole' in the graph at $$x = 1$$?

Accepted Answer

Because the numerator and denominator are both zero at $$x = 1.$$

Question 9

Given the function $$f(x) = |x|/x$$, what is $$\lim_{x 	o 0^+} f(x)$$?

Accepted Answer

$$1$$

Question 10

Given the function $$f(x) = |x|/x$$, what is $$\lim_{x 	o 0^-} f(x)$$?

Accepted Answer

$$-1$$

Question 11

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

Accepted Answer

$$2$$

Question 12

Examine the function $$\cos(1/x) as$$ $$x$$ approaches $$0. $$What can you say about $$\lim_{x 	o 0} \cos(1/x)$$?

Accepted Answer

The limit does not exist because the function oscillates between $$-1$$ and $$1.$$

Question 13

A patient’s body temperature $$T(t) is $$modeled by a function that is undefined at $$t = 5$$ hours due to a measurement error. If $$\lim_{t 	o 5} T(t) = 98.6$$, what does this mean?

Accepted Answer

The temperature approaches $$98.6 as$$ $$t$$ approaches $$5$$, regardless of the value at $$t = 5.$$

Question 14

Suppose $$f(x) is $$defined for all $$x$$ except $$x = 3$$, and $$\lim_{x 	o 3} f(x) = 7. $$Which of the following is necessarily true?

Accepted Answer

$$f(x)$$ approaches $$7 as$$ $$x$$ approaches $$3$$

Question 15

A chemical reaction rate $$R(c) is $$modeled by a function with $$\lim_{c 	o 0} R(c) = 5. $$What does this tell you about the reaction rate as the concentration $$c$$ approaches zero?

Accepted Answer

The reaction rate approaches $$5 as$$ $$c$$ approaches $$0.$$

Chapter 2: Limits – Definition and Evaluation

Study Guide - Practice Questions

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