Determining the Limit of a Rational Function | Understanding and Techniques

Limit of a rational function r(x) = f(x)/g(x)

The limit of a rational function r(x) = f(x)/g(x) as x approaches a certain value, say c, is the value that r(x) approaches as x gets arbitrarily close to c

The limit of a rational function r(x) = f(x)/g(x) as x approaches a certain value, say c, is the value that r(x) approaches as x gets arbitrarily close to c. It is denoted as:

lim┬(x→c)⁡〖(f(x)/g(x))〗 or lim┬(x→c)⁡f(x)/g(x)

To find the limit of a rational function, you need to analyze the behavior of the numerator f(x) and the denominator g(x) as x approaches the given value c.

If f(x) and g(x) are both continuous functions, meaning they have no breaks or jumps in their graphs as x varies, then you can simply evaluate r(c) to find the limit.

However, there are cases where f(x) and/or g(x) may have some discontinuities, such as vertical asymptotes or removable discontinuities. In such cases, you will need to apply additional techniques to find the limit.

Here are some common scenarios for finding the limit of a rational function:

1. If both f(x) and g(x) are polynomials, you can evaluate the limit simply by substituting the given value c into the function. For example, if f(x) = x^2 + 3x – 2 and g(x) = 2x – 1, finding lim┬(x→2)⁡ (f(x)/g(x)) would involve substituting x = 2 into the function.

2. If the degree of the numerator f(x) is less than the degree of the denominator g(x), then the limit as x approaches ∞ (infinity) or -∞ (minus infinity) is 0. For example, lim┬(x→∞)⁡ (3x^2 + 2x)/(5x^3 + 7x^2 + 1) = 0.

3. If the degree of the numerator f(x) is equal to the degree of the denominator g(x), then the limit as x approaches ∞ or -∞ is given by the ratio of the leading coefficients. For example, lim┬(x→-∞)⁡ (2x^3 – 5x^2 + 3)/(7x^3 + 2x^2 – 8) = -2/7.

4. If f(x) and/or g(x) have vertical asymptotes or removable discontinuities, you may need to factor out common factors, cancel out common terms, or use techniques like L’Hôpital’s Rule to simplify the function before evaluating the limit. For example, if f(x) = x^2 – 4 and g(x) = x – 2, finding lim┬(x→2)⁡ (f(x)/g(x)) would involve canceling out the common factor of x – 2 and evaluating the simplified function.

It’s important to note that these are general guidelines, and there are more advanced techniques and cases that may require further analysis. Overall, finding the limit of a rational function involves understanding the algebraic behavior of the function and applying the appropriate strategies to evaluate it.

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