Q: Which of the following statements about continuity is true?

A function is continuous at $$x=a if$$ $$\lim_{x \to a} f(x)$$ exists and equals $$f(a).$$

Q: Which of the following is a correct application of the squeeze theorem?

If $$-\sqrt{x} \leq \sqrt{x} \sin \left( \frac{1}{x} \right) \leq \sqrt{x}$$ and $$\lim_{x \to 0^+} \sqrt{x} = 0$$, then $$\lim_{x \to 0^+} \sqrt{x} \sin \left( \frac{1}{x} \right) = 0.$$

Question 1

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

Accepted Answer

$$10$$

Question 2

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

Accepted Answer

The limit does not exist.

Question 3

Evaluate $$\lim_{x 	o 0} \left( \frac{7x - x}{x} ight).$$

Accepted Answer

$$6$$

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 $$\lim_{x 	o 3^-} f(x).$$

Accepted Answer

$$5$$

Question 5

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

Accepted Answer

The limit does not exist.

Question 6

Find $$\lim_{x 	o 1} \left( \frac{x^2$ - 2}{x^2$ + x} ight).$$

Accepted Answer

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

Question 7

For the function $$f(x) = \begin{cases} 3x-2, & x\u003c2 \ x^2$, & 2 \leq x \u003c 3 \ x-4, & 3 \leq x \end{cases}$$, describe the set of $$x-$$values where $$f(x) is $$continuous using interval notation.

Accepted Answer

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

Question 8

Identify any points of discontinuity for the function $$f(x) = \begin{cases} 3x-2, & x\u003c2 \ x^2$, & 2 \leq x \u003c 3 \ x-4, & 3 \leq x \end{cases}$$ and explain if the discontinuity is removable.

Accepted Answer

Discontinuities at $$x=2$$ and $$x=3$$; neither is removable.

Question 9

Use the limiting value of the slope of secants through the point $$P(2,6) to $$find the slope of the graph of $$f(x) at $$that point, where $$f(x) = 2x^2$ - 3x + 6.$$

Accepted Answer

Question 10

What is the equation of the line tangent to the graph of $$f(x) = 2x^2$ - 3x + 6 at $$the point $$P(2,6)$$?

Accepted Answer

$$y = 5x - 4$$

Question 11

Evaluate $$\lim_{x 	o 0^+} \sqrt{x} \sin \left( \frac{1}{x} ight)$$ using the squeeze theorem.

Accepted Answer

$$0$$

Question 12

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$. What is $$\lim_{t 	o \infty} C(t)$$ and what does it mean in a medical context?

Accepted Answer

$$0$$; the medication is completely eliminated over time.

Question 13

A car's position is given by $$s(t) = t^3$ - 4t^2$ + 6t. $$Find the instantaneous velocity at $$t=2.$$

Accepted Answer

$$-4$$

Question 14

Which of the following statements about continuity is true?

Accepted Answer

A function is continuous at $$x=a if$$ $$\lim_{x 	o a} f(x)$$ exists and equals $$f(a).$$

Question 15

If $$\lim_{x 	o a} f(x) = L$$ and $$\lim_{x 	o a} g(x) = M$$, what is $$\lim_{x 	o a} [f(x) + g(x)]$$?

Accepted Answer

$$L + M$$

Question 16

A chemical reaction rate is modeled by $$r(t) = 10 - 2t. $$What is $$\lim_{t 	o 5} r(t)$$?

Accepted Answer

$$0$$

Question 17

Which of the following is a correct application of the squeeze theorem?

Accepted Answer

If $$-\sqrt{x} \leq \sqrt{x} \sin \left( \frac{1}{x} ight) \leq \sqrt{x}$$ and $$\lim_{x 	o 0^+} \sqrt{x} = 0$$, then $$\lim_{x 	o 0^+} \sqrt{x} \sin \left( \frac{1}{x} ight) = 0.$$

Question 18

Suppose $$\lim_{x 	o 0} \frac{\sin x}{x} = 1. $$What does this imply about the behavior of $$\sin x$$ near $$x=0$$?

Accepted Answer

$$\sin x is $$approximately equal to $$x$$ for small $$x.$$

Limits, Continuity, and Tangents: Calculus I Study Notes

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