Domains and Limits in Calculus: Rational, Radical, and Composite Functions

Domains and Limits in Calculus

Domains of Rational Functions

The domain of a function is the set of all real numbers for which the function is defined. For rational functions (functions of the form $f(x) = \frac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials), the domain excludes any value of $x$ that makes the denominator zero.

• Key Point: To find the domain, set the denominator equal to zero and solve for $x$. Exclude these values from the domain.

• Example: For $f(x) = \frac{x^2-9}{x^2+2x-3}$, set $x^2+2x-3=0$ and solve for $x$.

• Solution: $x^2+2x-3=0 \implies (x+3)(x-1)=0 \implies x=-3, 1$

• Domain: All real numbers except $x=-3$ and $x=1$.

Domains of Radical Functions

For radical functions involving even roots (such as square roots), the expression inside the root (the radicand) must be non-negative for the function to be defined over the real numbers.

• Key Point: Set the radicand greater than or equal to zero and solve for $x$.

• Example: For $f(x) = \sqrt{2x-4}$, set $2x-4 \geq 0$.

• Solution: $2x-4 \geq 0 \implies x \geq 2$

• Domain: $[2, \infty)$

Domains of Composite Functions

A composite function $f(g(x))$ is defined only when $x$ is in the domain of $g$ and $g(x)$ is in the domain of $f$.

• Key Point: Find the domain of the inner function $g(x)$, then ensure that $g(x)$ produces values in the domain of $f$.

• Example: If $f(x) = 2x^2-3x+7$ and $g(x) = 2x-5$, then $f(g(x)) = 2(2x-5)^2 - 3(2x-5) + 7$.

• Domain: Since both $f$ and $g$ are polynomials, their domains are all real numbers.

Limits and Indeterminate Forms

The limit of a function as $x$ approaches a value describes the behavior of the function near that value. If direct substitution leads to an indeterminate form (such as $\frac{0}{0}$), algebraic manipulation or special limit laws are used.

• Key Point: If $\lim_{x \to a} f(x)$ results in $\frac{0}{0}$, factor, rationalize, or use L'Hôpital's Rule if appropriate.

• Example: $\lim_{x \to 1} \frac{x^2-1}{x-1}$

• Solution: Factor numerator: $x^2-1 = (x-1)(x+1)$. Cancel $(x-1)$: $\lim_{x \to 1} x+1 = 2$.

Special Trigonometric Limits

Some limits involving trigonometric functions are fundamental in calculus, especially as $x$ approaches $0$.

• Key Point: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ and $\lim_{x \to 0} \frac{1-\cos x}{x} = 0$

• Example: $\lim_{x \to 0} \frac{\tan(3x)}{3x} = 1$

• Application: These limits are used to evaluate more complex limits involving trigonometric expressions.

Table: Common Indeterminate Forms and Strategies

Indeterminate Form	Example	Strategy
$\frac{0}{0}$	$\lim_{x \to 2} \frac{x^2-4}{x-2}$	Factor and cancel common terms
$\frac{\infty}{\infty}$	$\lim_{x \to \infty} \frac{2x^2+1}{x^2-3}$	Divide numerator and denominator by highest power of $x$
$0 \cdot \infty$	$\lim_{x \to 0^+} x \ln x$	Rewrite as a quotient and use L'Hôpital's Rule
$\infty - \infty$	$\lim_{x \to \infty} (\sqrt{x^2+x} - x)$	Rationalize or combine terms

Summary of Steps for Finding Domains and Limits

• For rational functions, exclude values that make the denominator zero.

• For radical functions with even roots, set the radicand $\geq 0$.

• For composite functions, ensure the output of the inner function is in the domain of the outer function.

• For limits, substitute first; if indeterminate, manipulate algebraically or use limit laws.

• For trigonometric limits, use standard results and identities.

Additional info: The above notes synthesize the worksheet's focus on domains (rational, radical, composite) and limits (including indeterminate forms and trigonometric limits), providing definitions, examples, and a summary table for common strategies.