Unit 1 Limits: How Calculus Describes Approaching, Not Just Plugging In

Limit

The value that a function’s outputs approach as the input gets close to a particular number (even if the function is not well-behaved or not defined at that number).

Limit Notation

The statement (\lim_{x\to a} f(x)=L) meaning the outputs of (f(x)) get arbitrarily close to (L) when (x) is taken sufficiently close to (a).

“Approaches” Language

A verbal/arrow form of a limit statement: (f(x)\to L) as (x\to a), equivalent to (\lim_{x\to a} f(x)=L).

Two-Sided Limit

A limit (\lim_{x\to a} f(x)) that considers (x) approaching (a) from both the left and the right.

Left-Hand Limit

(\lim_{x\to a^-} f(x)=L): the values of (f(x)) approach (L) as (x) approaches (a) using inputs less than (a).

Right-Hand Limit

(\lim_{x\to a^+} f(x)=L): the values of (f(x)) approach (L) as (x) approaches (a) using inputs greater than (a).

Condition for a Two-Sided Limit to Exist

(\lim{x\to a} f(x)) exists iff both one-sided limits exist and are equal: (\lim{x\to a^-} f(x)=\lim_{x\to a^+} f(x)).

Limits vs. Function Values

The limit (\lim_{x\to a} f(x)) can exist even if (f(a)) is different from the limit or if (f(a)) does not exist, because limits depend on behavior near (a), not at (a).

Removable Discontinuity (Hole)

A break in a graph where a single point is missing (often from a factor that cancels); the limit at that (x)-value may exist even though the function value is undefined or different.

Jump Discontinuity

A discontinuity where the left-hand and right-hand limits exist but are different, so the two-sided limit does not exist.

Graphical Cue: Open Circle

A graph symbol indicating a “hole” (the function is not defined at that point on the curve), though the limit may still be the (y)-value approached there.

Graphical Cue: Filled Dot

A graph symbol indicating the actual function value (f(a)) at that input (a), which may differ from the limit.

Vertical Asymptote (Limit Behavior)

A vertical line (x=a) the graph approaches where (f(x)) grows without bound (toward (\infty) or (-\infty)) as (x\to a).

Estimating a Limit from a Graph

Find the (y)-value the curve approaches as (x) approaches (a) from the left and right; if both sides approach the same (y), that is the limit.

Estimating a Limit from a Table

Use values of (x) closer and closer to (a) from below and above and see whether (f(x)) settles toward a single number; compare both sides.

Direct Substitution (When Valid)

A method where (\lim_{x\to a} f(x)=f(a)) if (f) is continuous at (a) (e.g., polynomials, and rational functions with nonzero denominator at (a)).

Limit Laws (Algebraic Properties)

Rules that allow limits of sums, differences, constant multiples, products, and quotients (when denominator limit is nonzero) to be computed from limits of simpler pieces.

Quotient Law Requirement

For (\lim{x\to a} \frac{f(x)}{g(x)}=\frac{L}{M}), the limit (M=\lim{x\to a} g(x)) must exist and satisfy (M\neq 0).

Indeterminate Form (\frac{0}{0})

A substitution result that does not determine the limit by itself; different expressions producing (0/0) can have different limits, so algebraic manipulation is needed.

Factoring and Canceling (Limit Technique)

Rewrite an expression by factoring numerator/denominator and canceling common factors (for (x\neq a)) to remove a (0/0) form and then compute the limit.

Conjugate (Rationalizing)

For an expression with radicals (e.g., (\sqrt{1+x}-1)), multiplying by the conjugate ((\sqrt{1+x}+1)) can simplify and eliminate (0/0).

Complex Fraction Simplification

A technique for limits with fractions inside fractions: combine terms over a common denominator or multiply by the least common denominator to expose factors that cancel.

Special Trigonometric Limit

The foundational limit (\lim_{x\to 0} \frac{\sin x}{x}=1), used to evaluate many trig limits by rewriting into this form.

Squeeze Theorem

If (g(x)\le f(x)\le h(x)) near (a) and (\lim{x\to a} g(x)=\lim{x\to a} h(x)=L), then (\lim_{x\to a} f(x)=L).

Bounding with Trig Inequalities

Using facts like (-1\le \sin x\le 1) and (-1\le \cos x\le 1) (often multiplied by (x^2) or (|x|)) to set up the Squeeze Theorem, especially for oscillatory expressions such as (\sin(1/x)) or (\cos(1/x)).