Welcome to the Chapter 2 Review!

Hello Math Mavericks! Get ready to conquer Chapter 2 with this review. We will be covering key concepts such as limits, continuity, and rates of change. Let's get started!

Understanding Limits

One of the fundamental concepts in calculus is the limit. The limit of a function $f(x)$ as $x$ approaches a value $c$ is denoted as:

$$ \lim_{x \to c} f(x) = L $$

This means that as $x$ gets arbitrarily close to $c$, the value of $f(x)$ gets arbitrarily close to $L$.

One-Sided Limits

  • Left-Hand Limit: $ \lim_{x \to c^-} f(x) $ (approaching $c$ from the left)
  • Right-Hand Limit: $ \lim_{x \to c^+} f(x) $ (approaching $c$ from the right)

For a limit to exist at $x = c$, both the left-hand limit and the right-hand limit must exist and be equal.

Example:

Consider the piecewise function:

$$ f(x) = \begin{cases} \frac{x}{2-x} & \text{if } x<3 \\ (x-1)^2 & \text{if } x>3 \end{cases} $$

Then we have:

  • $ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} \frac{x}{2-x} = -3 $
  • $ \lim_{x \to 3^+} f(x) = \lim_{x \to 3^+} (x-1)^2 = 4 $
  • Since the left-hand limit and the right-hand limit are not equal, $ \lim_{x \to 3} f(x) $ does not exist (DNE).

Continuity

A function $f(x)$ is continuous at a point $x = c$ if the following three conditions are met:

  1. $f(c)$ is defined.
  2. $ \lim_{x \to c} f(x) $ exists.
  3. $ \lim_{x \to c} f(x) = f(c) $

If any of these conditions are not met, the function is discontinuous at $x = c$.

Example:

Consider the function $f(x) = 2 - \frac{4}{x}$. This function is not continuous at $x = 0$ because it is not defined at $x = 0$.

Average and Instantaneous Rate of Change

The average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by:

$$ \frac{f(b) - f(a)}{b - a} $$

This is the slope of the secant line passing through the points $(a, f(a))$ and $(b, f(b))$.

The instantaneous rate of change of a function $f(x)$ at a point $x = c$ is given by the derivative $f'(c)$, which can be found by taking the limit of the average rate of change as the interval approaches zero:

$$ f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} $$

The instantaneous rate of change represents the slope of the tangent line to the graph of $f(x)$ at $x = c$.

Example:

Consider the function $f(x) = (x-2)^2$. To find the average rate of change over the interval $[2.9, 3]$, we calculate:

$$ \frac{f(3) - f(2.9)}{3 - 2.9} = \frac{1 - 0.01}{0.1} = \frac{0.99}{0.1} = 9.9 $$

Limits at Infinity

Limits at infinity describe the behavior of a function as $x$ approaches positive or negative infinity.

  • $ \lim_{x \to \infty} f(x) = L $ means that as $x$ becomes very large, $f(x)$ approaches $L$.
  • $ \lim_{x \to -\infty} f(x) = L $ means that as $x$ becomes very small (large negative), $f(x)$ approaches $L$.

Example:

Consider the function $h(x) = \frac{2x-1}{x+1}$.

  • $ \lim_{x \to \infty} \frac{2x-1}{x+1} = 2 $
  • $ \lim_{x \to -\infty} \frac{2x-1}{x+1} = 2 $

This indicates the horizontal asymptote is $y = 2$

Keep Practicing!

Remember, practice makes perfect! Work through plenty of problems, and don't hesitate to ask questions. You've got this!