Mastering Calculus: How To Find The Limit Of A Function As X Approaches Infinity

limits as x approaches infinity

horizontal asymptotes

In calculus, finding the limit of a function as x approaches infinity is an important concept. It helps us determine the behavior of a function as x increases without bound. Here are some key points to remember when finding the limit as x approaches infinity:

1. If the degree of the numerator is smaller than the degree of the denominator, the limit is zero. For example, the limit of 3x^2 / (4x^3 + 2) as x approaches infinity is zero.

2. If the degree of the numerator is the same as the degree of the denominator, the limit is the ratio of their leading coefficients. For example, the limit of (6x^3 + 4x^2 – 2) / (3x^3 + 2x^2 – 1) as x approaches infinity is 6/3, which simplifies to 2.

3. If the degree of the numerator is larger than the degree of the denominator, the limit is either infinity or negative infinity, depending on the signs of the leading coefficients. For example, the limit of (4x^4 + 6x^3 – 5x + 2) / (2x^3 – 3x^2 + 1) as x approaches infinity is positive infinity.

4. If the function has a horizontal or slant asymptote, the limit is the equation of the asymptote. For example, the limit of (3x^2 + 2x – 1) / (x – 5) as x approaches infinity is the equation of the slant asymptote, which is y = 3x + 17.

In summary, finding the limit of a function as x approaches infinity involves examining the degree and signs of the terms, and determining if there is a horizontal or slant asymptote. By applying these rules, we can accurately determine the behavior of a function as x increases without bound.

More Answers: