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Tony's Intertwingled Memex • Reference Manual

Limits

Formulas & Definitions
Core Definition

The value that a function approaches as the input approaches a target point. Formally, for a function f(x)f(x) defined on an open interval around cc (except possibly at cc itself), we write lim⁡x→cf(x)=L\lim_{x \to c} f(x) = L if for every ϵ>0\epsilon > 0 there exists a δ>0\delta > 0 such that if 0<∣x−c∣<δ0 < |x - c| < \delta, then ∣f(x)−L∣<ϵ|f(x) - L| < \epsilon.

Concept Explanation

Limits analyze the behavior of a function near a point rather than at that point. This is crucial for evaluating functions at coordinates where they are algebraically undefined (such as 00\frac{0}{0} indeterminate forms). In higher dimensions (Calculus III), a limit exists only if the function approaches the same value LL along every possible path approaching cc.

Visual / Geometric Intuition

Interactive Desmos Graph

Engineering Applications

Engineers use limits to model transient responses in electrical circuits, determine critical stability limits in control systems, and resolve aerodynamic singularities in fluid simulation software near boundary layers.

Example Problem

Evaluate the limit analytically:

lim⁡x→3x2−9x−3\lim_{x \to 3} \frac{x^2 - 9}{x - 3}

Solution

Factor the numerator as a difference of squares:

x2−9x−3=(x−3)(x+3)x−3\frac{x^2 - 9}{x - 3} = \frac{(x - 3)(x + 3)}{x - 3}

Since a limit evaluates behavior as x→3x \to 3 but x≠3x \neq 3, we can cancel the common factor (x−3)(x - 3):

lim⁡x→3(x−3)(x+3)x−3=lim⁡x→3(x+3)\lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3} (x + 3)

Now, substitute x=3x = 3 directly:

3+3=63 + 3 = 6

Connections & References