Q: A function $$f(x) is $$continuous at $$x = a if$$:

$$\lim_{x \to a} f(x)$$ exists, $$f(a) is $$defined, and $$\lim_{x \to a} f(x) = f(a)$$

Q: Which of the following best describes the Squeeze Theorem?

If $$f(x) \leq g(x) \leq h(x)$$ and $$\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$$, then $$\lim_{x \to a} g(x) = L$$

Question 1

Which of the following statements best describes the concept of a limit as $$x$$ approaches $$a$$?

Accepted Answer

The value that $$f(x)$$ approaches as $$x$$ gets arbitrarily close to $$a.$$

Question 2

Evaluate the following limit analytically: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$2$$

Question 3

Which of the following is NOT a valid method for evaluating limits?

Accepted Answer

Ignoring discontinuities

Question 4

A function $$f(x) is $$continuous at $$x = a if$$:

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists, $$f(a) is $$defined, and $$\lim_{x 	o a} f(x) = f(a)$$

Question 5

Given $$f(x) = \frac{1}{x-3}$$, what type of discontinuity occurs at $$x = 3$$?

Accepted Answer

Infinite (essential) discontinuity

Question 6

Which of the following best describes the Squeeze Theorem?

Accepted Answer

If $$f(x) \leq g(x) \leq h(x)$$ and $$\lim_{x 	o a} f(x) = \lim_{x 	o a} h(x) = L$$, then $$\lim_{x 	o a} g(x) = L$$

Question 7

Suppose $$f(x) is $$defined as follows: $$f(x) = \begin{cases} x^2$ & x \u003c 1 \ 2x - 1 & x \geq 1 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

Yes, because both one-sided limits and $$f(1)$$ are equal

Question 8

Which of the following is an example of a removable discontinuity?

Accepted Answer

$$f(x) = \frac{x^2$ - 1}{x - 1} at$$ $$x = 1$$

Question 9

A patient’s medication concentration $$C(t) in $$the bloodstream is modeled by $$C(t) = \frac{10}{t+2}. $$What is $$\lim_{t 	o 0} C(t)$$ and what does it represent?

Accepted Answer

$$5$$, the initial concentration immediately after administration

Question 10

Which of the following statements about the Intermediate Value Theorem is correct?

Accepted Answer

If $$f(x) is $$continuous on $$[a, b]$$ and $$N is $$between $$f(a)$$ and $$f(b)$$, then there exists $$c in$$ $$[a, b]$$ such that $$f(c) = N$$

Question 11

Evaluate $$\lim_{x 	o 0} \frac{\sin(x)}{x}.$$

Accepted Answer

$$1$$

Question 12

A car’s velocity is given by $$v(t) = 3t^2$ - 2t + 1. $$What is the instantaneous velocity at $$t = 2$$?

Accepted Answer

$$9$$

Question 13

Which of the following is the precise definition of $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that if $$0 \u003c |x - a| \u003c \delta$$, then $$|f(x) - L| \u003c \epsilon$$

Question 14

If $$\lim_{x 	o 3^-} f(x) = 2$$ and $$\lim_{x 	o 3^+} f(x) = 5$$, what can you conclude about $$\lim_{x 	o 3} f(x)$$?

Accepted Answer

The limit does not exist

Question 15

Which of the following is a necessary condition for a function to be continuous at a point $$x = a$$?

Accepted Answer

All of the above

Question 16

A chemical reaction’s rate is modeled by $$R(t) = e$$^{-t}$$. What is $$\lim_{t 	o \infty} R(t)$$ and what does it represent?

Accepted Answer

$$0$$, the reaction rate approaches zero as time increases

MAT 2410 Unit 1 Calculus Exam Study Guide & Review Problems

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