Q: Which of the following is an example of a jump discontinuity?

A function where $$\lim\limits_{x \to 1^-} f(x) \neq \lim\limits_{x \to 1^+} f(x).$$

Q: Suppose $$f(x) is $$continuous at $$x = a. $$Which of the following must be true?

$$\lim\limits_{x \to a} f(x)$$ exists and equals $$f(a)$$

Q: Which of the following statements about the Squeeze Theorem is correct?

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

Question 1

Given $$f(x) = x^2$ + 2x$$, use the definition of the derivative to find the slope of the tangent line at $$x = 3.$$

Accepted Answer

$$4$$

Question 2

Which of the following best describes a removable discontinuity in a function's graph?

Accepted Answer

A hole in the graph where the limit exists but the function is not defined or defined differently.

Question 3

Suppose $$f(x) = \begin{cases} x^2$ + 1 & x \geq 2 \ 7 - x & x \u003c 2 \end{cases}. $$Does $$f(x)$$ have a limit as $$x$$ approaches $$2$$?

Accepted Answer

Yes, and the limit is $$5.$$

Question 4

Evaluate $$\lim\limits_{x 	o 4} \frac{x^2$ - 16}{x - 4}.$$

Accepted Answer

$$8$$

Question 5

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

Accepted Answer

A function where $$\lim\limits_{x 	o 1^-} f(x) 
eq \lim\limits_{x 	o 1^+} f(x).$$

Question 6

Use the Squeeze Theorem to evaluate $$\lim\limits_{x 	o 0} x^2$ \cos\left(\frac{1}{x}ight).$$

Accepted Answer

$$0$$

Question 7

Find the equation of the tangent line to $$f(x) = x^2$ - 3x at$$ $$x = -2.$$

Accepted Answer

$$y = -7x - 18$$

Question 8

For the function $$f(x) = \frac{x^2$ - 1}{x - 1}$$, what type of discontinuity is present at $$x = 1$$?

Accepted Answer

Removable discontinuity

Question 9

Evaluate $$\lim\limits_{t 	o 0} \frac{3t}{t^2$ + 1}.$$

Accepted Answer

$$0$$

Question 10

A function $$f(x) is $$defined as $$f(x) = \sqrt{x+1}. $$What is $$\lim\limits_{x 	o 3} f(x)$$?

Accepted Answer

Question 11

Suppose $$f(x) is $$continuous at $$x = a. $$Which of the following must be true?

Accepted Answer

$$\lim\limits_{x 	o a} f(x)$$ exists and equals $$f(a)$$

Question 12

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 13

If $$f(x) = \frac{1}{x-2}$$, what type of discontinuity does $$f(x)$$ have at $$x = 2$$?

Accepted Answer

Infinite discontinuity

Question 14

Given $$f(x) = \sqrt{x+1}$$, find $$\lim\limits_{x 	o -1^+} f(x).$$

Accepted Answer

$$0$$

Question 15

Which of the following statements about the Squeeze Theorem is correct?

Accepted Answer

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

Question 16

Evaluate $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 17

A patient’s medication concentration in the bloodstream is modeled by $$C(t) = \frac{3t}{t^2$ + 1}. $$What is the long-term concentration as $$t 	o \infty$$?

Accepted Answer

$$0$$

Question 18

Find the value of $$k so $$that $$f(x) = \begin{cases} x^2$ + k & x \geq 2 \ 7 - x & x \u003c 2 \end{cases} is $$continuous everywhere.

Accepted Answer

$$k = 3$$

Calculus I: Limits, Continuity, and the Definition of the Derivative

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