Welcome to the Chapter 2 Review for Calculus 1!

This chapter delves into the foundational concepts of limits and continuity. Understanding these concepts is crucial for mastering calculus. Let's begin by reviewing the key topics covered.

Limits

The concept of a limit is the bedrock of calculus. It describes the behavior of a function as the input approaches a certain value.

  • Definition: The limit of $f(x)$ as $x$ approaches $a$ is $L$, written as $\lim_{x \to a} f(x) = L$, if we can make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ to be sufficiently close to $a$, but not equal to $a$.
  • One-Sided Limits: We also consider one-sided limits: $\lim_{x \to a^-} f(x)$ (left-hand limit) and $\lim_{x \to a^+} f(x)$ (right-hand limit). For the limit to exist, both one-sided limits must exist and be equal.

Here are some examples of finding limits. From the provided materials, we see examples such as:

  • $\lim_{x \to 0} sin(x^3 + 3x) = sin(0) = 0$
  • $\lim_{x \to \infty} \frac{6t^2 + t - 5}{9 - 2t^2} = \frac{6}{-2} = -3$ (by dividing by the highest power of $t$)
  • $\lim_{x \to \infty} \frac{\sqrt{2x^2 + 1}}{3x - 5} = \frac{\sqrt{2}}{3}$ (after dividing by $x$ and simplifying)
  • $\lim_{x \to -\infty} (x^2 + 2x^7) = -\infty$

Sometimes, direct substitution leads to indeterminate forms, and we need to use algebraic manipulation or other techniques. For example:

  • $\lim_{x \to 2} \frac{2 - x}{\sqrt{x + 2} - 2} = -4$ (by multiplying by the conjugate)

Continuity

A function is continuous at a point if its limit at that point exists, the function is defined at that point, and the limit equals the function's value.

  • Definition: A function $f(x)$ is continuous at $x = a$ if:
    1. $f(a)$ is defined.
    2. $\lim_{x \to a} f(x)$ exists.
    3. $\lim_{x \to a} f(x) = f(a)$.

If any of these conditions fail, the function is discontinuous at $x = a$. The attached notes show examples of piecewise functions and their limits. Consider the example:

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

In this case, we need to check the limits at the breakpoints (x=1 and x=2) to determine continuity.

Tangent Lines

The concept of the derivative, which you'll explore later, is closely tied to the tangent line. The slope of the tangent line to the curve $y = f(x)$ at the point $(a, f(a))$ is given by the limit:

$$m = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Or, equivalently:

$$m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$

Once you find the slope $m$, you can use the point-slope form of a line, $y - f(a) = m(x - a)$, to find the equation of the tangent line.

Example: Find the tangent line to $y = \frac{2}{7 - x}$ at the point $(5, 1)$. Using the first formula from the attached notes, the slope $m = \frac{1}{2}$. Thus, the equation of the tangent line is $y = \frac{1}{2}x - \frac{3}{2}$.

Keep Practicing!

Reviewing these concepts and practicing example problems will solidify your understanding of Chapter 2. Good luck with your studies!