AP calculus ab

1.1 Introducing Limits: Graphical and Numerical Approaches

• Definition of a Limit:

  • The limit of $f(x)$ as $x$ approaches $a$ is $L$, written as $<br>\lim_{x \to a} f(x) = L$, if the values of $f(x)$ get arbitrarily close to $L$ as $x$ gets closer to $a$ from both sides.

  • The function does not need to be defined at $x = a$ for the limit to exist.

• One-Sided Limits:

  • Left-hand limit: $\lim_{x \to a^-} f(x)$ represents the value $f(x)$ approaches as $x$ approaches $a$ from values less than $a$.

  • Right-hand limit: $\lim_{x \to a^+} f(x)$ represents the value $f(x)$ approaches as $x$ approaches $a$ from values greater than $a$.

• Existence of a Limit:

  • A two-sided limit exists if and only if both one-sided limits exist and are equal:
    $\lim<em>{x \to a} f(x) = L \iff \lim</em>{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$

1.2 Algebraic Properties and Evaluation of Limits

• Direct Substitution:

  • For continuous functions, evaluate $\lim_{x \to a} f(x)$ by substituting $x = a$ directly into $f(x)$.

• Indeterminate Form $\frac{0}{0}$:

  • If direct substitution yields $\frac{0}{0}$, algebraic manipulation is required:

  • Factoring: Factor the numerator and denominator to cancel common factors causing the zero.

  • Rationalization: Multiply numerator and denominator by the conjugate of an expression containing a radical.

  • Simplifying Complex Fractions: Find a common denominator to simplify fractions within fractions.

1.3 Infinite Limits and Limits at Infinity

• Vertical Asymptotes (Infinite Limits):

  • If $\lim<em>{x \to a^-} f(x) = \pm\infty$ or $\lim</em>{x \to a^+} f(x) = \pm\infty$, then the line $x = a$ is a vertical asymptote of $f(x)$.

• Horizontal Asymptotes (Limits at Infinity):

  • Evaluating $\lim<em>{x \to \infty} f(x) = L$ or $\lim</em>{x \to -\infty} f(x) = L$ determines horizontal asymptotes at $y = L$.

  • For rational functions $\frac{P(x)}{Q(x)}$:

  • If degree of $P(x)$ < degree of $Q(x)$, the limit is $0$.

  • If degree of $P(x)$ = degree of $Q(x)$, the limit is the ratio of leading coefficients.

  • If degree of $P(x)$ > degree of $Q(x)$, the limit is $\infty$ or $-\infty$.

1.4 Continuity and Discontinuity

• Definition of Continuity at a Point:

  • A function $f(x)$ is continuous at $x = c$ if and only if:

  1. $f(c)$ is defined.

  2. $\lim_{x \to c} f(x)$ exists.

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

• Types of Discontinuities:

  • Removable Discontinuity (Hole): $\lim<em>{x \to c} f(x)$ exists, but $\lim</em>{x \to c} f(x) \neq f(c)$ or $f(c)$ is undefined.

  • Jump Discontinuity: $\lim<em>{x \to c^-} f(x)$ and $\lim</em>{x \to c^+} f(x)$ exist, but $\lim<em>{x \to c^-} f(x) \neq \lim</em>{x \to c^+} f(x)$.

  • Infinite Discontinuity: One or both one-sided limits approach $\infty$ or $-\infty$.

1.5 The Intermediate Value Theorem (IVT)

• Statement of IVT:

  • If $f(x)$ is continuous on the closed interval $[a, b]$, and $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one value $c$ in the open interval $(a, b)$ such that $f(c) = N$.

• Application Steps:

  1. State that $f(x)$ is continuous on $[a, b]$.

  2. Evaluate $f(a)$ and $f(b)$.

  3. Confirm that $N$ lies between $f(a)$ and $f(b)$.

  4. Conclude that there exists $c \in (a, b)$ such that $f(c) = N$.