Q: Which of the following is the definition of the derivative of $$f(x) at$$ $$x=a$$?

$$\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: Suppose a function $$h(x)$$ has a jump discontinuity at $$x=2. $$Which of the following is true?

$$\lim\limits_{x \to 2^-} h(x) \neq \lim\limits_{x \to 2^+} h(x)$$

Question 1

Evaluate the following limit: $$\lim\limits_{x 	o 2} (2x^2$ - 3x + 6)$$

Accepted Answer

$$8$$

Question 2

Find $$\lim\limits_{x 	o 0} \left( \sin \left( \frac{5}{x-5} ight) ight).$$

Accepted Answer

$$0$$

Question 3

Evaluate $$\lim\limits_{x 	o 0} \left( \frac{\frac{1}{x} - x}{x} ight).$$

Accepted Answer

$$\infty$$

Question 4

Given the piecewise function $$f(x) = \begin{cases} 3x-2, & x\u003c2 \ x^2$, & 2 \leq x \u003c 3 \ x-4, & 3 \leq x \end{cases}$$, find $$\lim\limits_{x 	o 3^-} f(x).$$

Accepted Answer

$$9$$

Question 5

Evaluate $$\lim\limits_{x 	o 1} \left( \frac{x}{x^2$ - x} ight).$$

Accepted Answer

$$\infty$$

Question 6

Evaluate $$\lim\limits_{x 	o \infty} \left( \frac{x^2$ - 2}{x^2$ + x} ight).$$

Accepted Answer

$$1$$

Question 7

For which values of $$x is $$the function $$f(x) = \begin{cases} 3x-2, & x\u003c2 \ x^2$, & 2 \leq x \u003c 3 \ x-4, & 3 \leq x \end{cases}$$ continuous? Express your answer in interval notation.

Accepted Answer

Question 8

A function $$f(x) is $$defined as $$f(x) = \frac{x^2$ - 4}{x-2}. $$What is $$\lim\limits_{x 	o 2} f(x)$$?

Accepted Answer

$$2$$

Question 9

Suppose $$f(x) is $$continuous everywhere except at $$x=1$$ and $$x=3. $$Which of the following statements is true?

Accepted Answer

$$f(x) is $$discontinuous at $$x=1$$ and $$x=3$$

Question 10

Given $$\lim\limits_{x 	o 0} \sqrt{x} \sin \left( \frac{1}{x} ight)$$, which of the following is true?

Accepted Answer

The limit is $$0$$

Question 11

A car's position at time $$t is $$given by $$s(t) = t^2$ + 2t. $$What is the instantaneous velocity at $$t=3$$?

Accepted Answer

$$10$$

Question 12

Find the equation of the tangent line to $$f(x) = x^2$ at the point (2$$,4)$$.

Accepted Answer

$$y = 4x - 4$$

Question 13

Which of the following best describes a removable discontinuity?

Accepted Answer

A point where the function is not defined but the limit exists

Question 14

If $$\lim\limits_{x 	o a^-} f(x) = \lim\limits_{x 	o a^+} f(x) = L$$ but $$f(a) 
eq L$$, what type of discontinuity does $$f$$ have at $$x=a$$?

Accepted Answer

Removable discontinuity

Question 15

A chemical reaction's concentration is modeled by $$C(t) = 5e$$^{-0.2t}$$. What is the instantaneous rate of change of concentration at t=0$?

Accepted Answer

$$-1$$

Question 16

Which of the following is the definition of the derivative of $$f(x) at$$ $$x=a$$?

Accepted Answer

$$\lim\limits_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 17

A function $$g(x) is $$defined as $$g(x) = \frac{1}{x}$$ for $$x 
eq 0. $$Which of the following statements is true?

Accepted Answer

$$g(x) is $$discontinuous at $$x=0$$

Question 18

Suppose a function $$h(x)$$ has a jump discontinuity at $$x=2. $$Which of the following is true?

Accepted Answer

$$\lim\limits_{x 	o 2^-} h(x) 
eq \lim\limits_{x 	o 2^+} h(x)$$

Question 19

A patient’s temperature $$T(t) in $$degrees Celsius is modeled by $$T(t) = 37 + 2e$$^{-0.5t}$$. What is the rate of change of temperature at t=0$?

Accepted Answer

$$-1$$

Calculus I: Limits, Continuity, and Tangent Lines – Study Notes

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