Study Notes on Functions and Limits

The domain of functions typically includes concepts such as:
- Fractions: Must consider the denominator cannot equal zero, which might restrict the domain.
- Even Roots: The expression inside must be non-negative (≥ 0).
- Logarithms: The argument for a logarithm must be greater than zero (> 0).

As the denominator of a fraction grows large, the value of the fraction approaches zero. This highlights the relationship between the denominator and the overall value of a fraction.
- Example: $\frac{1}{n}$ approaches 0 as $n$ approaches infinity.

Conversely, when discussing the numerator of a fraction, as the numerator grows large, the entire fraction goes to infinity.
- For instance, in the fraction $\frac{n}{1}$, if $n$ approaches infinity, then the fraction itself approaches infinity.

Notation such as $\infty$ indicates unbounded growth. When evaluating limits:
- For a function like $f(x)$ as $x$ approaches infinity, limits can be expressed and evaluated, noting the behavior (e.g., $f(x) \to \infty$).

Given a polynomial function $f(x)$, for example:$
- $f(x) = \frac{x^3 + 100}{x^2 + x + 3}$,
Analysis of polynomials involves looking at the highest degree term in both the numerator and the denominator to determine end behavior. As $x \to \infty$, the behavior can be expressed as dominant term analysis:
- If the bottom degree (denominator) becomes larger than the top degree (numerator), $f(x) \to 0$.

Case Example: $f(x) = \frac{5x^3 + x^2 + x + 15}{3x^2 + 25x + 100}$.
- Evaluate the dominant term: as $x \to \infty$, it is found that the fraction approaches 5 (leading coefficient of 5 from numerator).

If $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials:
- If $\deg[Q(x)] < \deg[P(x)]$, then $f(x) \to \infty$.
- If $\deg[Q(x)] = \deg[P(x)]$, then $f(x) \to \frac{lead\ p}{lead\ q}$ as $x \to \infty$.

For rational functions such as: $f(x) = \frac{2x^3 + 5x^2 - x + 5}{x^3 + 1000}$:
- Evaluating as $x\to\infty$ will yield a limit corresponding to the leading coefficients of the polynomial degrees.
- Example: Here it evaluates to $2$.

Y-Intercept Calculation: $f(0)$ gives the y-intercept.
- For example: if $f(0)=2$, then the y-intercept is at (0,2).
X-Intercept Calculation: Set $f(x)=0$ to find x-values.
- In a function like $f(x) = x + 1$, setting this to zero gives the x-intercept x=-1.
Vertical Asymptotes: Defined as where the function approaches infinity, typically found by setting the denominator to zero and solving.
- Example: Denominator results such as $(x+3)=0$ leads to $x=-2$ being a vertical asymptote.
Horizontal Asymptotes: Consider end behavior to assess these limits; compare degrees of leading terms to explore behavior as $x \to \infty$.
- Evaluating limits as $x = \infty$, for various function forms derive limits to find asymptotes.

Example Function: Review values such as $f(-4)= -16$, $f(2)= 32.6$.
- This helps to demonstrate function behavior across different inputs.
Functions may also be evaluated for critical points and behavior, including limits and asymptotic analysis.