Determine end behaviour of a function

End behaviour describes how the output $f(x)$ changes as the input $x$ increases or decreases without bound, expressed with words, graphs, tables, or limits such as $x\to\infty$ and $x\to-\infty$ . For common polynomial, rational, exponential, and logarithmic functions, it is determined by features such as the polynomial’s leading term, dominant numerator and denominator degrees, or growth and decay, while remaining distinct from behaviour near a finite endpoint or vertical asymptote; formal epsilon-based limits and more advanced function classes are not included.

Detailed Explanation: Determine end behaviour of a function

End behaviour describes what happens to $(f(x))$ when $x$ becomes very large positive or very large negative.

For a polynomial, the leading term determines the end behaviour. The leading term is the term with the highest power of $x$ .

Worked example

Determine the end behaviour of

f(x)=-2x^3+5x^2-x+7.

Step 1: Identify the leading term.

The leading term is

-2x^3.

The lower-degree terms become less important when $(|x|)$ is very large.

Step 2: Examine the right end, when $(x\to\infty)$ .

As $(x\to\infty)$ ,

x^3\to\infty.

Because the coefficient is negative,

-2x^3\to-\infty.

Therefore,

\boxed{\lim_{x\to\infty}f(x)=-\infty}.

In words: As $x$ increases without bound, $(f(x))$ decreases without bound.

Step 3: Examine the left end, when $(x\to-\infty)$ .

As $(x\to-\infty)$ ,

x^3\to-\infty.

Multiplying by $(-2)$ changes the direction:

-2x^3\to\infty.

Therefore,

\boxed{\lim_{x\to-\infty}f(x)=\infty}.

In words: As $x$ decreases without bound, $(f(x))$ increases without bound.

So the graph rises on the far left and falls on the far right.

Learn by doing: Determine end behaviour of a function

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Function End Behaviour (Polynomials) - Function to Behaviour

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