How To Find The Limit At Infinity

Introduction to Limits at Infinity00:00:01

The video starts by explaining how to find the limit of a function as x approaches infinity. For example, for x squared, as x becomes very large, x squared also becomes infinity. Similarly, as x approaches negative infinity, x squared results in positive infinity.

Limits of Polynomial Functions00:01:42

For polynomial functions, when finding limits at infinity, only the term with the highest degree is significant. For instance, in 5 + 2x - x cubed, the limit as x approaches negative infinity is determined solely by -x cubed, leading to positive infinity. Another example analyzes -5x to the fourth as x approaches negative infinity, resulting in negative infinity.

Limits of Rational Functions: Bottom Heavy00:03:41

The video then covers rational functions. It explains that for 1/x, as x approaches infinity, the limit is zero because the denominator grows infinitely large. This concept applies generally: if the degree of the denominator is greater than the numerator (bottom heavy), the limit as x approaches infinity is always zero. This is demonstrated using 5x + 2 / 7x - x squared.

Limits of Rational Functions: Same Degree00:06:11

When the degree of the numerator and denominator are the same, the limit as x approaches infinity is the ratio of their leading coefficients. For example, the limit of (8x squared - 5x) / (4x squared + 7) as x approaches infinity is 8/4, which simplifies to 2. This is shown by dividing both numerator and denominator by the highest power of x.

Limits of Rational Functions: Top Heavy00:09:14

If the degree of the numerator is greater than the denominator (top heavy), the insignificant terms can be removed to simplify the expression. For (5x + 6x squared) / (3x - 8) as x approaches infinity, it simplifies to (6x squared) / (3x) or 2x, leading to positive infinity. Another example, with 5 + 2x - 3x cubed / 4x squared + 9x - 7, also demonstrates this simplification.