Unit 11 test: Introduction to Limits

The table shows values of $f$ near $x = 1$.

Estimate $\displaystyle \lim_{x \to 1} f(x)$.

The graph of $f$ is shown. Which statement is true?

Let $f(x) = \begin{cases} x + 4, & x < -1 \\ x^2 + 1, & x \ge -1. \end{cases}$ What is $\displaystyle \lim_{x \to -1} f(x)$?

Suppose $\displaystyle \lim_{x \to a} f(x) = 5$ and $\displaystyle \lim_{x \to a} g(x) = 2$. Find $\displaystyle \lim_{x \to a} \frac{f(x)\,g(x) + 3}{f(x) - g(x)}$.

Find $\displaystyle \lim_{x \to 4} \frac{x^2 - 16}{x - 4}$.

Find $\displaystyle \lim_{x \to 3} \frac{x^2 - 2x - 3}{x^2 - 9}$.

Find $\displaystyle \lim_{x \to 0} \frac{\sqrt{4 + x} - 2}{x}$.

What kind of discontinuity does $f(x) = \dfrac{x^2 - 1}{x - 1}$ have at $x = 1$?

List every $x$-value where $g(x) = \dfrac{x - 1}{x^2 - 3x + 2}$ is discontinuous.

The function $g(x) = \dfrac{x - 1}{x^2 - 3x + 2}$ from the previous problem has a removable discontinuity. What value should be assigned to $g$ at that point to make it continuous there?

Find $k$ so that $f(x) = \begin{cases} kx - 3, & x < 2 \\ x^2 + k, & x \ge 2 \end{cases}$ is continuous everywhere.

Let $f(x) = x^3 - 3x - 1$. On which interval does the Intermediate Value Theorem guarantee a zero of $f$?

Find the slope of the tangent line to $f(x) = x^2 - 4x$ at $x = 3$.

Find the equation of the tangent line to $f(x) = \dfrac{4}{x}$ at $x = 2$.

An object's position after $t$ seconds is $s(t) = 5t^2 + 2t$ meters. Find its instantaneous velocity at $t = 3$, in meters per second.