What is how to find horizontal asymptote?

Here's how to find horizontal asymptotes:

What is a Horizontal Asymptote?

A horizontal%20asymptote is a horizontal line that the graph of a function approaches as x tends to positive infinity (x → ∞) or negative infinity (x → -∞). It describes the end behavior of the function.

Finding Horizontal Asymptotes of Rational Functions

The primary method for finding horizontal asymptotes involves analyzing the degrees of the polynomials in the numerator and denominator of a rational%20function (a function that's a ratio of two polynomials). Let's say you have a rational function:

f(x) = P(x) / Q(x)

Where P(x) and Q(x) are polynomials.

Here are the rules:

• Case 1: Degree of P(x) < Degree of Q(x) If the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator, then the horizontal asymptote is y = 0 (the x-axis).

• Case 2: Degree of P(x) = Degree of Q(x) If the degree of the polynomial in the numerator is equal to the degree of the polynomial in the denominator, then the horizontal asymptote is y = a/b, where a is the leading coefficient of P(x) and b is the leading coefficient of Q(x). Divide the leading coefficients.

• Case 3: Degree of P(x) > Degree of Q(x) If the degree of the polynomial in the numerator is greater than the degree of the polynomial in the denominator, then there is no horizontal asymptote. There may be a slant (oblique) asymptote instead.

Beyond Rational Functions:

For functions that aren't rational functions (e.g., exponential functions, trigonometric functions, functions involving radicals), you'll generally need to evaluate the limits:

• lim (x→∞) f(x)
• lim (x→-∞) f(x)

If either of these limits exists and equals a finite number L, then y = L is a horizontal%20asymptote.

Important Considerations:

• A function can cross a horizontal asymptote. The asymptote describes the behavior as x approaches infinity, not necessarily the function's behavior for finite values of x.
• A function can have at most two horizontal asymptotes: one as x approaches positive infinity, and another (possibly the same one) as x approaches negative infinity.