Q: For $$f(x) = \frac{x^2$ - 2x}{x-2}$$ and $$g(x) = x^2$, compare $$\lim\limits_{x \to 2} f(x)$$ and $$\lim\limits_{x \to 2} g(x)$$. What do you observe?

$$\lim\limits_{x \to 2} f(x) = 2$$ and $$\lim\limits_{x \to 2} g(x) = 4$$

Question 1

Suppose $$f(x) = x^2$. What is $$\lim\limits_{x 	o 2} f(x)$$?

Accepted Answer

$$4$$

Question 2

In the notation $$\lim\limits_{x 	o 2^-} f(x) = 4$$, what does the negative sign above the 2 indicate?

Accepted Answer

The limit is taken as $$x$$ approaches 2 from values less than 2.

Question 3

Given $$g(x) = \frac{x^2$ - 2x}{x-2}$$, what is $$\lim\limits_{x 	o 2} g(x)$$?

Accepted Answer

$$2$$

Question 4

Which of the following is NOT a valid way to estimate a limit?

Accepted Answer

Using only the value at the point itself

Question 5

For the piecewise function $$h(x) = \begin{cases} x^2$, & x 
eq 2 \ 1, & x = 2 \end{cases}$$, what is $$\lim\limits_{x 	o 2} h(x)$$?

Accepted Answer

$$4$$

Question 6

If $$\lim\limits_{x 	o a^-} f(x) = L$$ and $$\lim\limits_{x 	o a^+} f(x) = M$$ with $$L 
eq M$$, what can be said about $$\lim\limits_{x 	o a} f(x)$$?

Accepted Answer

The limit does not exist

Question 7

Suppose $$\lim\limits_{x 	o 3} f(x) = 2$$ and $$\lim\limits_{x 	o 3} g(x) = 5. $$What is $$\lim\limits_{x 	o 3} [f(x) + g(x)]$$?

Accepted Answer

$$7$$

Question 8

Evaluate $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$2$$

Question 9

If $$\lim\limits_{x 	o 2} f(x) = 3$$, what is $$\lim\limits_{x 	o 2} 5f(x)$$?

Accepted Answer

$$15$$

Question 10

Suppose $$\lim\limits_{x 	o 4} f(x) = 2$$ and $$\lim\limits_{x 	o 4} g(x) = 3. $$What is $$\lim\limits_{x 	o 4} [f(x) \cdot g(x)]$$?

Accepted Answer

$$6$$

Question 11

Evaluate $$\lim\limits_{x 	o 0} \frac{1}{x}.$$

Accepted Answer

Does not exist

Question 12

For $$f(x) = \frac{3x+2}{2x^2$-4}$$, what is $$\lim\limits_{x 	o 2} f(x)$$?

Accepted Answer

Does not exist

Question 13

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

Accepted Answer

$$L - M$$

Question 14

Evaluate $$\lim\limits_{x 	o \infty} \frac{1}{x}.$$

Accepted Answer

$$0$$

Question 15

A pond has oxygen concentration $$f(t) = \frac{12t^2$ + 15t + 12}{t^2$ + t + 1}. $$What is $$\lim\limits_{t 	o \infty} f(t)$$?

Accepted Answer

$$12$$

Question 16

For $$f(x) = \frac{1}{x^2$}$$, what is $$\lim\limits_{x 	o \infty} f(x)$$?

Accepted Answer

$$0$$

Question 17

If $$\lim\limits_{x 	o a} f(x) = L$$ and $$k is a $$constant, what is $$\lim\limits_{x 	o a} [k \cdot f(x)]$$?

Accepted Answer

$$k \cdot L$$

Question 18

Suppose $$\lim\limits_{x 	o 2} f(x) = 3$$ and $$\lim\limits_{x 	o 2} g(x) = 4. $$What is $$\lim\limits_{x 	o 2} [f(x$$)^{g(x)}$$]$$?

Accepted Answer

$$81$$

Question 19

Evaluate $$\lim\limits_{x 	o 1} \log_2$ x.$$

Accepted Answer

$$0$$

Question 20

For $$f(x) = \frac{x^2$ - 2x}{x-2}$$ and $$g(x) = x^2$, compare $$\lim\limits_{x 	o 2} f(x)$$ and $$\lim\limits_{x 	o 2} g(x)$$. What do you observe?

Accepted Answer

$$\lim\limits_{x 	o 2} f(x) = 2$$ and $$\lim\limits_{x 	o 2} g(x) = 4$$

Limits and Their Properties in Business Calculus

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