Q: Which of the following statements about limits is TRUE?

If $$\lim\limits_{x \to a} f(x) = L$$, then for every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that $$|f(x) - L| \u003c \epsilon$$ whenever $$0 \u003c |x - a| \u003c \delta.$$

Question 1

Which of the following intervals correctly represents the domain of the function $$f(x) = \sqrt{x^2$ - 25x}$$?

Accepted Answer

$$x \in (-\infty, 0] \cup [25, \infty)$$

Question 2

Given $$f(x) = 2x - 1$$ and $$g(x) = 3x^2$ + 1$$, what is the composite function $$f(g(x))$$?

Accepted Answer

$$f(g(x)) = 2(3x^2$ + 1) - 1$$

Question 3

Find the limit: $$\lim\limits_{x 	o 7} \frac{x-7}{7x}$$

Accepted Answer

$$0$$

Question 4

Evaluate $$\lim\limits_{x 	o 3} \frac{49x - x^2$}{x-3}.$$

Accepted Answer

$$20$$

Question 5

Which of the following is the correct analytical simplification for $$\lim\limits_{x 	o 3} \frac{\sqrt{x^2$ - 1} - 2}{x-3}$$?

Accepted Answer

Question 6

Find $$\lim\limits_{x 	o 0} \frac{\sin(2x) \sec(x)}{x}.$$

Accepted Answer

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

Question 7

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin(x^2$)}{x \cos(x)}.$$

Accepted Answer

$$0$$

Question 8

Given $$\lim\limits_{x 	o 5} \frac{x^2$ - 25}{x-5}$$, what is the value of the limit?

Accepted Answer

$$10$$

Question 9

Suppose $$f(x) = x^3$ + x^2$ - 9x - 9. $$According to the Intermediate Value Theorem, why does $$f(x)$$ have at least one solution in $$[-5, 5]$$?

Accepted Answer

Because $$f(x) is $$continuous on $$[-5, 5]$$ and $$f(-5)$$ and $$f(5)$$ have opposite signs.

Question 10

Using the formal definition of the limit, for $$\lim\limits_{x 	o 4} 7x - 8 = 20$$, what is the value of $$\epsilon if$$ $$|x-4| \u003c \frac{\epsilon}{7}$$?

Accepted Answer

$$\epsilon = 7|x-4|$$

Question 11

If $$f(x) = \sqrt{x^2$ - 25}$$, which of the following values of $$x is $$NOT in the domain of $$f(x)$$?

Accepted Answer

$$x = 3$$

Question 12

Given $$f(x) = 2x - 1$$ and $$g(x) = 3x^2$ + 1$$, what is $$f(g(2))$$?

Accepted Answer

$$f(g(2)) = 23$$

Question 13

Which of the following is a correct application of the Intermediate Value Theorem?

Accepted Answer

If $$f(a)  0$$ for a continuous function $$f on$$ $$[a, b]$$, then there exists $$c \in (a, b)$$ such that $$f(c) = 0.$$

Question 14

Evaluate $$\lim\limits_{x 	o 0} \frac{\cos(x) - 1}{x}.$$

Accepted Answer

$$0$$

Question 15

If $$f(x) = x^2$ - 25$$, what is the set of $$x$$ for which $$f(x) \geq 0$$?

Accepted Answer

$$x \in (-\infty, -5] \cup [5, \infty)$$

Question 16

Find the value of $$\lim\limits_{x 	o 5} \frac{x^2$ - 25}{x-5}$$ using factorization.

Accepted Answer

$$10$$

Question 17

Which of the following statements about limits is TRUE?

Accepted Answer

If $$\lim\limits_{x 	o a} f(x) = L$$, then for every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that $$|f(x) - L| \u003c \epsilon$$ whenever $$0 \u003c |x - a| \u003c \delta.$$

Question 18

Given $$\lim\limits_{x 	o 0} \frac{\sin(2x)}{x}$$, what is the value of the limit?

Accepted Answer

$$2$$

Question 19

If $$f(x) = x^3$ + x^2$ - 9x - 9 is $$continuous on $$[-5, 5]$$ and $$f(-5)  0$$, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is at least one $$c \in (-5, 5)$$ such that $$f(c) = 0.$$

Calculus Study Notes: Functions, Limits, and Theorems

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