Q: Which of the following is the correct definition of continuity for a function $$f(x) at$$ $$x=a$$ using limits?

$$\displaystyle \lim_{x \to a} f(x) = f(a)$$

Q: A function $$h(x)$$ has a jump discontinuity at $$x=3. $$Which of the following best describes this situation?

$$\displaystyle \lim_{x \to 3^-} h(x) \neq \lim_{x \to 3^+} h(x)$$

Question 1

Which of the following is the correct definition of continuity for a function $$f(x) at$$ $$x=a$$ using limits?

Accepted Answer

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

Question 2

Which of the following is NOT a type of discontinuity shown in the study materials?

Accepted Answer

Oscillating discontinuity

Question 3

A function $$f(x) is $$defined as $$f(x) = \dfrac{x^2$ - 4}{x - 2}. $$For which value of $$x is$$ $$f(x)$$ not continuous?

Accepted Answer

$$x = 2$$

Question 4

Given the piecewise function $$g(x) = \begin{cases} -2x + b, & x \u003c 1 \ \ln(x), & x \geq 1 \end{cases}$$, what value of $$b$$ makes $$g(x)$$ continuous at $$x=1$$?

Accepted Answer

$$b = 2 + \ln(1)$$

Question 5

For the function $$f(x) = \sqrt{x-3}$$, on which interval is $$f(x)$$ continuous?

Accepted Answer

$$[3, \infty)$$

Question 6

Which of the following statements is TRUE about the Intermediate Value Theorem (IVT)?

Accepted Answer

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

Question 7

Suppose $$f(x) is $$continuous on $$[1, 5]$$ and $$f(1) = -2$$, $$f(5) = 3. $$Which of the following values is guaranteed to be achieved by $$f(x)$$ for some $$x in$$ $$(1, 5) by $$the IVT?

Accepted Answer

$$0$$

Question 8

Which of the following functions is NOT continuous on its entire domain?

Accepted Answer

$$f(x) = \dfrac{1}{x}$$

Question 9

Given $$g(x) = \begin{cases} ax + 2b, & x \leq 0 \ x^2$ + 3a - b, & 0  2 \end{cases}$$, what values of $$a$$ and $$b$$ make $$g(x)$$ continuous everywhere?

Accepted Answer

$$a = 3,\ b = -5$$

Question 10

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

Accepted Answer

$$f(a) is $$defined

Question 11

A function $$h(x)$$ has a jump discontinuity at $$x=3. $$Which of the following best describes this situation?

Accepted Answer

$$\displaystyle \lim_{x 	o 3^-} h(x) 
eq \lim_{x 	o 3^+} h(x)$$

Question 12

Which of the following real-world scenarios best illustrates the Intermediate Value Theorem?

Accepted Answer

A car's speedometer reads 30 mph at 2 PM and 60 mph at 3 PM, so at some time between 2 and 3 PM, the speed was exactly 45 mph.

Question 13

For which value of $$x$$ does the function $$f(x) = \dfrac{1}{x-5}$$ have a discontinuity?

Accepted Answer

$$x = 5$$

Question 14

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

Accepted Answer

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

Question 15

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b)$$, what does the IVT guarantee?

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f(c) = 0$$

Continuity and the Intermediate Value Theorem

Guida di studio - Domande di pratica

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