Local vs Global Behavior Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy

For

p(x) = x^4 - 100x^2 + 1

, which term dominates (a) near

x=0

and (b) for

|x| = 1000

Solution

1
Near $x=0$ : the constant term $1$ dominates since $x^4$ and $x^2$ are both tiny. $p(0)=1$ ; $p(0.01)\approx1-0.01=0.99$ (approximately $1$ ).
2
For $|x|=1000$ : $x^4=10^{12}$ vastly exceeds $100x^2=10^8$ . The $x^4$ term dominates. $p(1000)\approx10^{12}$ .

Answer

Near

x=0

: constant term

1

dominates; for

|x|=1000

x^4

term dominates

In polynomials, the dominant term depends on the scale: near the origin, lower-degree terms matter; for large

|x|

, the highest-degree term overwhelms all others.

About Local vs Global Behavior

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.

Learn more about Local vs Global Behavior →

More Local vs Global Behavior Examples

Example 1 easy

For [formula], describe: (a) local behavior near [formula] using the linear approximation, and (b) g

Example 2 medium

For [formula] (with [formula]), describe local behavior near [formula] and global behavior for large

Example 4 hard

The function [formula] appears bell-shaped. Describe its local behavior at [formula] (using the seco

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