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

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

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

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

Question 1

Given $$f(x) = 2x^2$ - 3x + 6$$, calculate $$\lim_{x 	o 2} f(x). $$What is the value?

Accepted Answer

$$8$$

Question 2

Evaluate $$\lim_{x 	o 0} \left( \frac{\sin\left(\frac{5}{x}ight)}{x} ight). $$Which of the following is correct?

Accepted Answer

The limit does not exist

Question 3

Find $$\lim_{x 	o 0} \left( \frac{\frac{1}{x} - x}{x} ight). $$What is the value?

Accepted Answer

$$\infty$$

Question 4

Let $$f(x) = \begin{cases} 3x-2, & x\u003c2 \ x^2$, & 2 \leq x \u003c 3 \ x-4, & 3 \leq x \end{cases}. $$What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 5

Evaluate $$\lim_{x 	o 1} \left( \frac{x}{x^2$-1} ight). $$What is the value?

Accepted Answer

Does not exist

Question 6

Find $$\lim_{x 	o 1} \left( \frac{x^2$-2}{x^3$+x} ight). $$What is the value?

Accepted Answer

$$-\frac{1}{2}$$

Question 7

For which intervals 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?

Accepted Answer

$$(-\infty, 2) \cup (2, 3) \cup (3, \infty)$$

Question 8

Which of the following points is a discontinuity for $$f(x) = \begin{cases} 3x-2, & x\u003c2 \ x^2$, & 2 \leq x \u003c 3 \ x-4, & 3 \leq x \end{cases}$$?

Accepted Answer

$$x=2$$ and $$x=3$$

Question 9

Is the discontinuity at $$x=2$$ for $$f(x)$$ removable? Why?

Accepted Answer

No, because the left and right limits are not equal

Question 10

Use the squeeze theorem and the fact that $$-\sqrt{x} \leq \sqrt{x} \sin\left(\frac{1}{x}ight) \leq \sqrt{x}$$ for $$x\u003e0 to $$evaluate $$\lim_{x 	o 0^+} \sqrt{x} \sin\left(\frac{1}{x}ight). $$What is the value?

Accepted Answer

$$0$$

Question 11

Given $$f(x) is $$continuous at $$x=a$$, which of the following must be true?

Accepted Answer

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

Question 12

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

Accepted Answer

$$6$$

Question 13

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

Accepted Answer

$$y = 4x - 4$$

Question 14

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

Accepted Answer

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

Question 15

If $$\lim_{x 	o a} f(x)$$ does not exist, which of the following is most likely true?

Accepted Answer

The left and right limits at $$x=a$$ are not equal

Question 16

Suppose $$f(x) is $$defined as $$f(x) = \frac{1}{x}$$ for $$x 
eq 0. $$What is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

Does not exist

Question 17

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

Accepted Answer

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

Question 18

A patient’s temperature $$T(t) is $$modeled by $$T(t) = 98.6 + 0.5t - 0.1t^2$. What is the rate of change of temperature at t=3$ hours?

Accepted Answer

$$0.2$$

Question 19

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

Accepted Answer

Jump discontinuity

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

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