Local vs Global Behavior Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Local vs Global Behavior.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Local behavior describes a function's properties near a specific point; global behavior describes its overall properties across the entire domain or as inputs grow without bound.

Local is "zoom in on one spot"; global is "zoom out to see the whole picture." Near x = 0, \sin(x) \approx x (local linear approximation), but globally it oscillates forever.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Local behavior is captured by limits and derivatives near a point; global behavior is captured by asymptotes, end behavior, and the function's overall shape.

Common stuck point: A function can behave completely differently at a local level than globally — a function might be increasing at a point while decreasing overall.

Sense of Study hint: Zoom in on the graph near the point of interest for local behavior, then zoom out to see the entire curve for global behavior. Compare both views.

Worked Examples

Example 1

easy
For f(x)=x^3-3x, describe: (a) local behavior near x=0 using the linear approximation, and (b) global behavior as x\to\pm\infty.

Solution

  1. 1
    Local at x=0: f'(x)=3x^2-3, f'(0)=-3. Linear approximation: f(x)\approx f(0)+f'(0)\cdot x = 0+(-3)x=-3x near x=0. Locally the function looks like a line with slope -3.
  2. 2
    Global behavior: for large |x|, the x^3 term dominates. f(x)\to+\infty as x\to+\infty; f(x)\to-\infty as x\to-\infty.
  3. 3
    Contrast: locally (near x=0) the function decreases; globally it grows without bound in both directions.

Answer

Local near x=0: f(x)\approx-3x (decreasing line); Global: f(x)\to\pm\infty with x^3 dominance
Local behavior captures the function's character near a specific point, often described by a tangent line. Global behavior describes what happens as x goes to infinity, usually governed by the highest-degree term.

Example 2

medium
For g(x) = \dfrac{\sin(x)}{x} (with g(0)=1), describe local behavior near x=0 and global behavior for large |x|.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
For p(x) = x^4 - 100x^2 + 1, which term dominates (a) near x=0 and (b) for |x| = 1000?

Example 2

hard
The function f(x) = e^{-x^2} appears bell-shaped. Describe its local behavior at x=0 (using the second-order Taylor expansion) and global behavior as x\to\pm\infty.
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Related Concepts

Function

Background Knowledge

These ideas may be useful before you work through the harder examples.

function definition