Math Core

This test covers the meaning of a limit, estimating limits from graphs and tables, the limit laws, algebraic techniques, the squeeze theorem, continuity and discontinuities, limits at infinity and asymptotes, and the intermediate value theorem.

Question 1

Let $f(x) = \begin{cases} 3 - x, & x < 1 \\ 4, & x = 1 \\ x^2 + 1, & x > 1. \end{cases}$ Find $\displaystyle \lim_{x \to 1} f(x)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 2

Which statement is always true?

Problems 3 and 4 use the graph of $T$ shown below.

The graph of T. Open circles at (0, 3) and (3, 2); filled dots at (0, 1) and (3, 0).Open in grapher →

Question 3

Find $\displaystyle \lim_{x \to 0} T(x) + T(0)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 4

Which statement about $T$ at $x = 3$ is true?

Question 5

The table shows selected values of a function $h$ .

$x$	$1.9$	$1.99$	$1.999$	$2.001$	$2.01$	$2.1$
$h(x)$	$1.25641$	$1.250627$	$1.250063$	$1.249938$	$1.249377$	$1.243902$

Based on the table, what is the best estimate of $\displaystyle \lim_{x \to 2} h(x)$ ? Give an exact value.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 6

Suppose $\displaystyle \lim_{x \to 1} f(x) = -2$ and $\displaystyle \lim_{x \to 1} g(x) = 4$ . Find $\displaystyle \lim_{x \to 1} \frac{f(x)^2 - 3g(x)}{f(x) + g(x)}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 7

Evaluate $\displaystyle \lim_{x \to 2} \frac{x^2 + 3x - 10}{x^2 - 4}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 8

Evaluate $\displaystyle \lim_{x \to 2} \frac{\sqrt{x + 7} - 3}{x - 2}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 9

What is $\displaystyle \lim_{x \to 0} \frac{\sin 4x}{3x}$ ?

Question 10

Suppose $2x \le f(x) \le x^2 + 1$ for all $x$ in the interval $(0, 2)$ . Find $\displaystyle \lim_{x \to 1} f(x)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 11

Find the value of $k$ that makes $f$ continuous for all real numbers.

f(x) = \begin{cases} kx + 1, & x \le 3 \\ kx^2 - 5, & x > 3 \end{cases}

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 12

Let $g(x) = \dfrac{x^2 - 5x + 4}{x^2 - 1}$ . Which statement is true?

Question 13

For the function $g(x) = \dfrac{x^2 - 5x + 4}{x^2 - 1}$ in the previous problem, what value should $g(1)$ be given so that $g$ is continuous at $x = 1$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 14

Evaluate $\displaystyle \lim_{x \to \infty} \frac{6x^2 - x}{3 - 2x^2}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 15

What is $\displaystyle \lim_{x \to -1^+} \frac{x - 2}{x + 1}$ ?

Question 16

The function $f$ is continuous on $[0, 6]$ , and selected values are shown.

$x$	$0$	$2$	$4$	$6$
$f(x)$	$2$	$-1$	$3$	$5$

What is the least number of zeros that $f$ must have on $[0, 6]$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.