$$f(a)$$ must be defined, $$\lim_{x \to a} f(x)$$ must exist, and $$\lim_{x \to a} f(x) = f(a)$$

No, because $$\lim_{x \to 0} f(x) \neq f(0)$$

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

$$Q(a) = 0$$ and $$P(a) \neq 0$$

Question 1

Evaluate $$\lim_{x 	o 1} \frac{1}{x-1} - \frac{1}{2(x+1)}. $$Which of the following is correct?

Accepted Answer

$$-\frac{1}{4}$$

Question 2

Given $$f(x) = \frac{x^2$ - x - 2}{x-2}$$, at which value of $$x is$$ $$f(x)$$ discontinuous?

Accepted Answer

$$x = 2$$

Question 3

Evaluate $$\lim_{x 	o \infty} \frac{2 - x^2$}{5x^2$ - x^2$}.$$

Accepted Answer

$$-\frac{1}{5}$$

Question 4

Which of the following statements is TRUE about the continuity of a function $$f at$$ $$x=a$$?

Accepted Answer

$$f(a)$$ must be defined, $$\lim_{x 	o a} f(x)$$ must exist, and $$\lim_{x 	o a} f(x) = f(a)$$

Question 5

Evaluate $$\lim_{x 	o 0} \left( \frac{1}{x} + \frac{5}{x(x-5)} ight). $$What is the result?

Accepted Answer

$$-1$$

Question 6

Given $$f(x) = \begin{cases} x^2$ & x 
eq 0 \ 1 & x = 0 \end{cases}$$, is $$f(x)$$ continuous at $$x=0$$?

Accepted Answer

No, because $$\lim_{x 	o 0} f(x) 
eq f(0)$$

Question 7

Evaluate $$\lim_{x 	o \infty} \frac{6x^4$ + 8x}{-2x^4$ + 5}. $$What is the result?

Accepted Answer

$$-3$$

Question 8

For polynomials $$P(x)$$ and $$Q(x) of $$degree $$m$$ and $$n$$ respectively, what is $$\lim_{x 	o \infty} \frac{P(x)}{Q(x)} if$$ $$m \u003c n$$?

Accepted Answer

$$0$$

Question 9

Evaluate $$\lim_{x 	o \infty} \frac{x$$^{12}$$ + 11x^4$}{4x^7$ - 2x^3$ - 9}. $$What is the result?

Accepted Answer

$$0$$

Question 10

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

Accepted Answer

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

Question 11

Evaluate $$\lim_{x 	o \infty} \frac{-x^3$ + 5x^2$ - 9}{4x^3$ - 5x^2$ + 9}. $$What is the result?

Accepted Answer

$$-\frac{1}{4}$$

Question 12

If $$f(x) = \frac{1}{x^2$}$$, what is $$\lim_{x 	o 0^+} f(x)$$?

Accepted Answer

$$+\infty$$

Question 13

A rational function $$R(x) = \frac{P(x)}{Q(x)} is $$discontinuous at $$x=a if$$:

Accepted Answer

$$Q(a) = 0$$ and $$P(a) 
eq 0$$

Question 14

Which of the following is the correct simplification for $$\frac{x^a}{x^b}$$?

Accepted Answer

$$x^{a-b}$$

Question 15

Evaluate $$\lim_{x 	o \infty} \frac{10x^3$ - 11x}{12x^6$ - 1}.$$

Accepted Answer

$$0$$

Question 16

If $$f(x) = \frac{1}{x^2$}$$, what is $$\lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$0$$

Question 17

A function $$f(x) is $$defined as $$f(x) = \left\lfloor x \right\rfloor. At $$which values of $$x is$$ $$f(x)$$ discontinuous?

Accepted Answer

At every integer value of $$x$$

Question 18

If $$P(x)$$ and $$Q(x)$$ are polynomials of degree $$m$$ and $$n$$ respectively, and $$m = n$$, what is $$\lim_{x 	o \infty} \frac{P(x)}{Q(x)}$$?

Accepted Answer

$$\frac{a_m}{b_n}$$

Question 19

Which of the following is NOT a requirement for a function to be continuous at a point $$x=a$$?

Accepted Answer

$$f(a) is $$not defined$

Limits at Infinity and Continuity of Functions: Study Notes

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