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

$$\lim_{x \to a} f(x) = f(a)$$ \, \text{for all} \, a \, \text{in the domain}$

$$\lim_{x \to 2^-} f(x) \neq \lim_{x \to 2^+} f(x)$$

$$\lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x)$$

Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

Horizontal Shift

Question 3

Given $$f(x) is $$continuous everywhere except at $$x = -7$$ and $$x = 5$$, which interval notation correctly describes where $$f(x) is $$continuous?

Accepted Answer

$$(-\infty, -7) \cup (-7, 5) \cup (5, \infty)$$

Question 4

For the function $$f(x) = \frac{x+8}{(x-8)(x-2)}$$, at which values of $$x$$ does $$f(x)$$ have vertical asymptotes?

Accepted Answer

$$x = 8$$ and $$x = 2$$

Question 5

For $$f(x) = \frac{x^2$}{x^2$ + 36}$$, on which interval is $$f(x)$$ continuous?

Accepted Answer

$$(-\infty, \infty)$$

Question 6

Consider $$f(x) = \sqrt{x-2}. $$What is the domain of $$f(x)$$, and where is it continuous?

Accepted Answer

$$[2, \infty)$$, continuous on $$[2, \infty)$$

Question 7

If $$g(x) = \sqrt{x+5}$$, what is the domain of $$g(x)$$?

Accepted Answer

$$[-5, \infty)$$

Question 8

For the piecewise function $$f(x) = \begin{cases} 3 & x  -1 \end{cases}$$, what is $$\lim_{x 	o -1} f(x)$$?

Accepted Answer

$$	ext{Does Not Exist}$$

Question 9

A function $$f(x) is $$defined as continuous on its domain. Which of the following statements is always true?

Accepted Answer

$$\lim_{x 	o a} f(x) = f(a)$$ \, 	ext{for all} \, a \, 	ext{in the domain}$

Question 10

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

Accepted Answer

$$\lim_{x 	o 2^-} f(x) 
eq \lim_{x 	o 2^+} f(x)$$

Question 11

If $$f(x) is $$continuous on $$(-\infty, -7) \cup (-7, 5) \cup (5, \infty)$$, which of the following statements is correct?

Accepted Answer

$$f(x) is $$discontinuous at $$x = -7$$ and $$x = 5$$

Question 12

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

Accepted Answer

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

Question 13

For $$f(x) = \sqrt{x-2}$$ and $$g(x) = \sqrt{x+5}$$, for which values of $$x$$ are both $$f(x)$$ and $$g(x)$$ continuous?

Accepted Answer

$$[2, \infty)$$ \cap [-5, \infty) = [2, \infty)$

Question 14

A function $$f(x) is $$defined as $$f(x) = \frac{x+8}{(x-8)(x-2)}. On $$which intervals is $$f(x)$$ continuous?

Accepted Answer

$$(-\infty, 2) \cup (2, 8) \cup (8, \infty)$$

Question 15

If $$f(x) is $$piecewise defined and has a jump at $$x = 1$$, which of the following is true about $$\lim_{x 	o 1} f(x)$$?

Accepted Answer

$$\lim_{x 	o 1^-} f(x) 
eq \lim_{x 	o 1^+} f(x)$$

Continuity and Types of Discontinuities in Functions

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