Limits of Polynomial and Rational Functions

Limit Evaluation of Polynomial Functions at Specific Real Numbers

When evaluating the limit of a polynomial function as $x$ approaches a real number $a$, the direct substitution property applies. Because polynomial functions are continuous everywhere on the real line, the limit as $x$ approaches $a$ is simply equal to the value of the function evaluated at $a$.

For example, to compute the limit of a quadratic function as $x$ approaches $2$:

$f(2) = \lim_{x \to 2}(x^2 + 3x - 2)$

$\lim_{x \to 2}(x^2 + 3x - 2) = (2)^2 + 3(2) - 2$

$\lim_{x \to 2}(x^2 + 3x - 2) = 4 + 6 - 2$

$\lim_{x \to 2}(x^2 + 3x - 2) = 8$

Similarly, evaluating another quadratic expression at $x = 2$ via direct substitution proceeds as follows:

$\lim_{x \to 2}(9x^2 + 2x - 4)$

$\lim_{x \to 2}(9x^2 + 2x - 4) = 9(2)^2 + 2(2) - 4$

$\lim_{x \to 2}(9x^2 + 2x - 4) = 9(4) + 4 - 4$

$\lim_{x \to 2}(9x^2 + 2x - 4) = 36$

Limits of Polynomial Functions at Infinity

To determine the limit of a polynomial function as $x$ approaches positive infinity ($+\infty$) or negative infinity ($-\infty$), only the highest-degree term (the dominant term) determines the end behavior of the polynomial. As $x$ becomes arbitrarily large in magnitude, lower-degree terms become negligible in comparison to the leading term.

Consider the limit of the quadratic polynomial as $x$ approaches positive infinity:

$\lim_{x \to +\infty}(3x^2 + 7x - 5)$

$\lim_{x \to +\infty}(3x^2 + 7x - 5) = \lim_{x \to +\infty}(3x^2)$

$\lim_{x \to +\infty}(3x^2) = +\infty$

Next, consider a quartic polynomial evaluated as $x$ approaches negative infinity:

$\lim_{x \to -\infty}(8x^4 + 6x^2 + 3)$

$\lim_{x \to -\infty}(8x^4 + 6x^2 + 3) = \lim_{x \to -\infty}(8x^4)$

Because the exponent $4$ is an even positive integer, $(-1)^4 = 1$, causing the expression to grow positively without bound:

$\lim_{x \to -\infty}(8x^4) = +\infty$

Limits of Rational Functions at Infinity

A rational function is a quotient of two polynomial functions. When evaluating the limit of a rational function as $x \to +\infty$ or $x \to -\infty$, the limit is governed by the ratio of the highest-degree term in the numerator to the highest-degree term in the denominator.

For the limit where the degree of the numerator is strictly greater than the degree of the denominator:

$\lim_{x \to +\infty}\left(\frac{x^5 - 2x^3 + 4}{x^3 - 2x + 1}\right)$

$\lim_{x \to +\infty}\left(\frac{x^5 - 2x^3 + 4}{x^3 - 2x + 1}\right) = \lim_{x \to +\infty}\left(\frac{x^5}{x^3}\right)$

$\lim_{x \to +\infty}\left(\frac{x^5}{x^3}\right) = \lim_{x \to +\infty}(x^2)$

$\lim_{x \to +\infty}(x^2) = +\infty$

For a polynomial ratio involving high-degree algebraic terms as $x$ approaches negative infinity:

$\lim_{x \to -\infty}\left(\frac{-x^7 + x^4 - 7x^2}{-x^6}\right)$

$\lim_{x \to -\infty}\left(\frac{-x^7 + x^4 - 7x^2}{-x^6}\right) = \lim_{x \to -\infty}\left(\frac{-x^7}{-x^6}\right)$

$\lim_{x \to -\infty}\left(\frac{-x^7}{-x^6}\right) = \lim_{x \to -\infty}(x)$

$\lim_{x \to -\infty}(x) = -\infty$